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Wize University Calculus 1 Textbook > Applications of Differentiation

1st and 2nd Derivative Tests

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Wize University Calculus 1 Textbook > Applications of Differentiation

Critical Points and Extrema

3 Activities

Wize University Calculus 1 Textbook > Limits

The Intermediate Value Theorem

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Wize University Calculus 1 Textbook > Limits

Continuity

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Let f(x)f(x) be a continuous function on the open interval (a,b)(a, b). Which of the following four statements are always true.

Select all that apply.
More 1st and 2nd Derivative Tests Questions:
Critical points and extrema
Find all local extreme values of the function f(x)=x3−6x2+9x+1f(x)=x^3-6x^2+9x+1
Derivative tests
Let f(x)=−x3+x2+xf(x) = -x^3+x^2+x and use the first derivative test to find and classify the extreme values.
Since f is a negative cubic and we’re not given an interval, f has no absolute max or min. Here, f’(x)=−3x2+2x+1f’(x)=-3x^2+2x+1, which exists everywhere. Let’s set it equal to zero and solve for x:
0=−3x2+2x+1=−(3x+1)(x−1)→x=1,−1/3.0 = -3x^2+2x+1 = -(3x+1)(x-1) \rightarrow x=1,-1/3.
Critical points and Extrema
Given this table of values, which of the following statements must NOT be true?
f(x)f′(x)f′′(x)x=0313x=100−1x=20−20\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}
Absolute Extrema on Closed Interval
Find the absolute maximum of f(x)=3x4−4x3\displaystyle f(x)=3x^{4}-4x^{3} on the interval [−1,2][-1,2] using the second derivative test.
Absolute Extrema on Closed Interval
Find the absolute extrema of f(x)=3x4−4x3\displaystyle f(x)=3x^{4}-4x^{3} on the interval [−1,2][-1,2] using the second derivative test.
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Critical Points and Extrema: Max and Min
Find and classify the critical points of f(x)=xex−exf(x)=xe^{x}-e^{x} as relative maxima or minima.
Practice: Absolute Extrema on Closed Interval
Q.\textbf{Q.} Find the absolute maximum and minimum values of f(x)=6x43−3x13\displaystyle f(x)=6x^{\frac{4}{3}}-3x^{\frac{1}{3}} on the interval [−1,1][-1,1] and then everywhere, using the second derivative test.
Absolute Extrema on Closed Interval
Find the absolute extrema of f(x)=3x4−4x3\displaystyle f(x)=3x^{4}-4x^{3} on the interval [−1,2][-1,2] using the second derivative test.
Derivative Tests
Which of the following five functions are concave up on their whole domain? Select all that apply.
Critical points and Extrema
Given this table of values, which of the following statements must NOT be true?
f(x)f′(x)f′′(x)x=0313x=100−1x=20−20\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Critical Points and Extrema
Let f(x)f(x) be a smooth function (a function that has infinitely many derivatives that exist everywhere) and consider x=ax = a . Which of the following statements are correct? Check off all correct statements.
Practice
f(x)=1x2+1\displaystyle f(x)=\frac{1}{x^2+1}.
The first and the second derivatives are
f′(x)=−2x(x2+1)2 and f′′(x)=6x2−2(x2+1)3.\boxed{f^{\prime}(x)=\frac{-2x}{\left(x^{2}+1\right)^{2}}\quad\text{ and }\quad f^{\prime\prime}(x)=\frac{6x^{2}-2}{\left(x^{2}+1\right)^3}.}
Practice
f(x)=1x2+1\displaystyle f(x)=\frac{1}{x^2+1}.
The first and the second derivatives are
f′(x)=−2x(x2+1)2 and f′′(x)=6x2−2(x2+1)3.\boxed{f^{\prime}(x)=\frac{-2x}{\left(x^{2}+1\right)^{2}}\quad\text{ and }\quad f^{\prime\prime}(x)=\frac{6x^{2}-2}{\left(x^{2}+1\right)^3}.}
Practice: Minimum Value (Similar to April 2018 Q45)
Find the minimum value of f(x)=14x4+13x3f(x)=\frac{1}{4}x^4+\frac{1}{3}x^3
Practice: Max, Min, and Inflection Points
Practice Question: Max, Min, and Inflection Points
Given the following table of values, which statement(s) must always be true about f(x)f\left(x\right)?
i.) f(x)f\left(x\right) has 3 critical points
Derivative tests
Let f(x)=−x3+x2+xf(x) = -x^3+x^2+x and use the first derivative test to find and classify the extreme values.
Since f is a negative cubic and we’re not given an interval, f has no absolute max or min. Here, f’(x)=−3x2+2x+1f’(x)=-3x^2+2x+1, which exists everywhere. Let’s set it equal to zero and solve for x:
0=−3x2+2x+1=−(3x+1)(x−1)→x=1,−1/3.0 = -3x^2+2x+1 = -(3x+1)(x-1) \rightarrow x=1,-1/3.
Critical points and extrema
Find all local extreme values of the function f(x)=x3−6x2+9x+1f(x)=x^3-6x^2+9x+1
Practice: Absolute Extrema on Closed Interval
Q.\textbf{Q.} Find the absolute maximum and minimum values of f(x)=6x43−3x13\displaystyle f(x)=6x^{\frac{4}{3}}-3x^{\frac{1}{3}} on the interval [−1,1][-1,1] and then everywhere, using the second derivative test.
Absolute Extrema on Closed Interval
Find the absolute extrema of f(x)=3x4−4x3\displaystyle f(x)=3x^{4}-4x^{3} on the interval [−1,2][-1,2] using the second derivative test.
Critical Points and Extrema: Max and Min
Find and classify the critical points of f(x)=xex−exf(x)=xe^{x}-e^{x} as relative maxima or minima.
Derivative Tests
Find the local extreme values of h(x)=x4−2x2h(x)=x^4-2x^2 using the second derivative test.
More Critical Points and Extrema Questions:
Critical points and extrema
Find all local extreme values of the function f(x)=x3−6x2+9x+1f(x)=x^3-6x^2+9x+1
Critical points and Extrema
For what values of pp the function f(x)=13x3+x2+px−1\displaystyle f(x)=\frac{1}{3}x^3+x^2+px-1 has at least one critical point.
Critical points and Extrema
Given this table of values, which of the following statements must NOT be true?
f(x)f′(x)f′′(x)x=0313x=100−1x=20−20\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}
Critical Points and Relative Extrema
Find all relative extrema for the function f(x)=4x−3x2/3f(x)=4x-3x^{2/3}.
Critical points and extrema
Where does the absolute maximum occur for the function f(x)=ex+1xf(x) = e^x + \frac{1}{x} on the interval [1,2][1, 2] .
Example: Absolute max and min
Example: Absolute max and minWize Concept Clarifier
Find the absolute extrema (max/min) of f(x)=e−x2+1f\left(x\right)=e^{-x^2}+1 on the interval [−1, ∞]\left[-1,\ \infty\right].
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Max, Min, and Inflection Points
Given the following table of values, which statement(s) must always be true about f(x)f\left(x\right)? (Assume f(x)f\left(x\right) is continuous)
Select all that apply.
Critical Points and Extrema: Max and Min
Find and classify the critical points of f(x)=xex−exf(x)=xe^{x}-e^{x} as relative maxima or minima.
Critical Points and Extrema
What can be said about the extreme values of even and odd functions?
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Critical Points and Relative Extrema
Find all relative extrema for the function f(x)=4x−3x2/3f(x)=4x-3x^{2/3}.
Critical Points and Extrema
The following is the graph of f′′(x)f''\left(x\right), which of the following statements (if any) is correct?
Critical points and Extrema
Given this table of values, which of the following statements must NOT be true?
f(x)f′(x)f′′(x)x=0313x=100−1x=20−20\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}
Critical Points and Extrema
What can be said about the extreme values of even and odd functions?
What are the critical numbers for the function f(x)=x3exf(x)=x^3e^x ?
Conside the function f(x)=x1/4(x2−81)f(x)=x^{1/4}(x^2-81) which has derivative f′(x)=9(x2−9)4x3/4f'(x)=\frac{9(x^2-9)}{4x^{3/4}}. Which of the following lists all the critical points of f(x)f(x)?
Practice: Minimum Value (Similar to April 2018 Q45)
Find the minimum value of f(x)=14x4+13x3f(x)=\frac{1}{4}x^4+\frac{1}{3}x^3
How many critical numbers does the function f(x)=x2exf(x)=x^2e^x have?
Practice: Max, Min, and Inflection Points
Practice Question: Max, Min, and Inflection Points
Given the following table of values, which statement(s) must always be true about f(x)f\left(x\right)?
i.) f(x)f\left(x\right) has 3 critical points
Concavity, Inflection Points, Extrema and Intervals
Given f(x)=2−2x+13x3f(x)=2-2x+\frac{1}{3}x^3, find
the intervals of increase and decrease
the local maximum and minimum values
Critical points and extrema
Find all local extreme values of the function f(x)=x3−6x2+9x+1f(x)=x^3-6x^2+9x+1
Critical Points and Extrema
The graph of the derivative f′(x)f^{'}(x) is shown
Where are the local minimums of f(x)f(x)located?
The graph of the derivative f′(x)f^{'}(x) is shown
Where are the local minimums of f(x)f(x)located? Also list the intervals for which f is concave up or down.
Find the critical points for f(x)=x−3x3f(x)=x-3\sqrt[3]{x}.
The graph of the derivative f′(x)f^{'}(x) is shown
Where are the local minimums of f(x)f(x)located? Also list the intervals for which f is concave up or down.
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Critical Points and Relative Extrema
Find all relative extrema for the function f(x)=4x−3x2/3f(x)=4x-3x^{2/3}.
Critical Points and Extrema: Max and Min
Find and classify the critical points of f(x)=xex−exf(x)=xe^{x}-e^{x} as relative maxima or minima.
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f(x)=x2−2ln⁡xf(x)=x^2-2\ln{x}
Critical points and Extrema
For what values of pp the function f(x)=13x3+x2+px−1\displaystyle f(x)=\frac{1}{3}x^3+x^2+px-1 has at least one critical point.
Consider the function f(x)=x2/3(x−5)f(x)=x^{2/3}(x-5) which has derivative f′(x)=5(x−2)3x1/3f'(x)=\frac{5(x-2)}{3x^{1/3}}. Which of the following lists all the critical points of f(x)f(x)?
Find the critical points for f(x)=x−3x3f(x)=x-3\sqrt[3]{x} as an ordered pair (x, y).
For the following function, determine all the local and global minimum/maximum and inflection points over the given interval
g(x)=x+2x on  (0,∞)g(x)=\sqrt{x}+\frac{2}{x} \text{ on }\ (0,\infty)
f′(x)f'(x) plot for a differentiable function is depicted below.
a) Find all critical point of ff in the range shown in the plot.
b) Determine which of the critical points found in the previous part are local max/min.
Find the critical points of f(x)=x+1x\displaystyle f(x)=x+\frac{1}{x}.
Find the critical point(s) of f(x)=x−3x3f(x)=x-3\sqrt[3]{x}.
Intervals of increase and decrease
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f(x)=x3ln⁡xf(x) = x^3 \ln x
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
y=e4x2xy=\frac{e^{4x}}{2x}
Which of the following statements is true about the function f(x)=(x2−3)e−xf\left(x\right)=\left(x^2-3\right)e^{-x}?
i.) The function has critical points at x=±3x=\pm\sqrt{3}
ii.) The function has a relative maximum at x=3x=3
Concavity and Inflection points
Give the graph of f′(x)f'\left(x\right) below, which of the following statments must be true about the graph of f(x)f\left(x\right)?
(Select all that apply)
Intervals of increase and decrease
If f′(x)=x5x+1(x−1)3(x+2)f'\left(x\right)=x^5\sqrt{x+1}\left(x-1\right)^3\left(x+2\right), which of the following statements about the graph of f(x)f\left(x\right) must be true?
(Select all that apply)
Example: Absolute max and min
Example: Absolute max and minWize Concept Clarifier
Find the absolute extrema (max/min) of f(x)=e−x2+1f\left(x\right)=e^{-x^2}+1 on the interval [−1, ∞]\left[-1,\ \infty\right].
Max, Min, and Inflection Points
Given the following table of values, which statement(s) must always be true about f(x)f\left(x\right)? (Assume f(x)f\left(x\right) is continuous)
Select all that apply.
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Where does the absolute maximum occur for the function f(x)=ex+1xf(x) = e^x + \frac{1}{x} on the interval [1,2][1, 2] .
Critical points and extrema
Where does the absolute maximum occur for the function f(x)=ex+1xf(x) = e^x + \frac{1}{x} on the interval [1,2][1, 2] .
Critical Points and Extrema
Let f(x)f(x) be a smooth function (a function that has infinitely many derivatives that exist everywhere) and consider x=ax = a . Which of the following statements are correct? Check off all correct statements.
Critical points and extrema
f′(x)f'(x) plot for a differentiable function is depicted as below. Answer following questions.
Determine which of the critical points are local max/min
Critical points and extrema
f′(x)f'(x) plot for a differentiable function is depicted as below. Answer following questions.
Find all critical point of ff in the range shown in the plot.
Critical Points and Extrema
A math-inclined skier is going off a jump and getting his photo taken. His height above the photographer is given by f(t)=−t2+6t+5f(t)=-t^2+6t+5, where ttis the time in seconds after the skier leaves the ground. When should the photographer take the photo to get the skier at his maximum height above ground?
Critical points and extrema: Maximum and minimum on closed intervals
Find the absolute maximum value of f(x)=2ex(sin⁡x−cos⁡x)f(x)=2e^x(\sin x-\cos x)
on the interval [π/2,π][\pi/2,\pi].
Critical points and extrema: Maximum and Minimum on closed intervals
Find the absolute minimum of f(x)=arcsin(x2)f(x)=\text{arcsin}\left(\frac{x}{2}\right) on the interval [1,2][1,2].
Critical Points and Extrema: Maximum and Minimum on Closed Intervals
Find the absolute maximum of value of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1}
on the interval [0,2][0,2].
More The Intermediate Value Theorem Questions:
The Intermediate Value Theorem
Show that there is at least one solution to the equation ln⁡x=esin⁡(πx2)\ln x=e^{\sin\left(\frac{\pi x}{2}\right)}.
The Intermediate Value Theorem
Use the Intermediate Value Theorem to show that the function f(x)=x3−4x+2f\left(x\right)=x^3-4x+2 crosses the x−x-axis in the interval [0,2][0,2] .
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
The Intermediate Value Theorem: At Least One Solution
Show that the function f(x)=x3−7x2+6x−13=0f(x)=x^3-7x^2+6x-13=0 must possess at least one real-valued solution.
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
Intermediate Value Theorem
If f(x)f\left(x\right) is a continuous function on [−1,1]\left[-1,1\right], such that f(−1)=−πf\left(-1\right)=-\pi and f(1)=πf\left(1\right)=\pi, which statement is always true?
The Intermediate Value Theorem
Show that there is at least one solution to the equation ln⁡x=esin⁡(πx2)\ln x=e^{\sin\left(\frac{\pi x}{2}\right)}.
Indetermiante Value Theorem
Does x3−1+sin⁡(x)=0x^3-1+\sin\left(x\right)=0 have a root in [0,1]\left[0,1\right]?
The Intermediate Value Theorem
Show that the equation sin⁡x=x2−x−1\sin x=x^2-x-1 has at least one root in the interval (0,3)(0,3), i.e., there exists a number c∈(0,3)c \in (0,3) such that sin⁡c=c2−c−1\sin c=c^2-c-1 .
The Intermediate Value Theorem
Use the Intermediate Value Theorem to show that the function f(x)=x3−4x+2f\left(x\right)=x^3-4x+2 crosses the x−x-axis in the interval [0,2][0,2] .
The Intermediate Value Theorem
Determine if the function f(x)=x3−6x+4f(x)=x^3-6x+4 has a root in the domain [0,1] [0,1] .
Does 3x2+cos⁡x=33x^2+\cos{x}=3 have a root in (0,1)(0,1)?
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
The Intermediate Value Theorem: At Least One Solution
Show that the function f(x)=x3−7x2+6x−13=0f(x)=x^3-7x^2+6x-13=0 must possess at least one real-valued solution.
Practice: IVT & Equation Solutions
Practice: IVT & Equation Solutions
Show that there is at least one solution to the equation ln⁡x=esin⁡(πx2)\ln x=e^{\sin\left(\frac{\pi x}{2}\right)}.
The Intermediate Value Theorem
Let f(x)f(x) be a function so that
f(x),f′(x),f′′(x)f(x), f'(x), f''(x) exist and are continuous for all xx, and
∣f(x)−sin⁡(x)∣≤1/4|f(x) - \sin(x)| \leq 1/4 for all xx.
Alex is thinking about tan⁡x\tan x function. He knows that tan⁡1>0\tan1>0 and tan⁡2<0\tan2<0 He concludes that tan⁡x\tan x has a root in (1, 2). Justify if he is right or wrong.
The intermediate value theorem
f(x)=15x5+43x3+3x−2f(x)=\frac{1}{5}x^5+\frac{4}{3}x^3+3x-2 has
f(x):[0,1]→[1,3]f(x):[0,1]\rightarrow[1,3] is continuous. Does f(x)f(x) intersect with y=2x+1y=2x+1 ?
final114
Megan was thinking about the function f(x)=3(x−3)3f(x)=\dfrac{3}{(x-3)^3} and she thought f(0)<0f(0)<0 and also f(4)>0f(4)>0 .So, by the Intermediate Value Theorem, this function has a root in the interval (0,4)(0,4). Is she right?
Indetermiante Value Theorem
Does x3−1+sin⁡(x)=0x^3-1+\sin\left(x\right)=0 have a root in [0,1]\left[0,1\right]?
The Intermediate Value Theorem
Determine if the function f(x)=x3−6x+4f(x)=x^3-6x+4 has a root in the domain [0,1] [0,1] .
Does 3x2+cos⁡x=33x^2+\cos{x}=3 have a root in (0,1)(0,1)?
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
The Intermediate Value Theorem: At Least One Solution
Show that the function f(x)=x3−7x2+6x−13=0f(x)=x^3-7x^2+6x-13=0 must possess at least one real-valued solution.
The Intermediate Value Theorem
Use the Intermediate Value Theorem to show that the function f(x)=x3−4x+2f\left(x\right)=x^3-4x+2 crosses the x−x-axis in the interval [0,2][0,2] .
The Intermediate Value Theorem
Show that the equation sin⁡x=x2−x−1\sin x=x^2-x-1 has at least one root in the interval (0,3)(0,3), i.e., there exists a number c∈(0,3)c \in (0,3) such that sin⁡c=c2−c−1\sin c=c^2-c-1 .
🦊 TRICKY! Megan Mah was thinking about the y=tan⁡xy=\tan x function. She thought tan (1) > 0 and also tan(2) < 0, so, by the Intermediate Value Theorem, this function has a root in the interval (1,2). Is she right?
Does x3−1+sin⁡(x)=0x^3-1+\sin\left(x\right)=0 have at least one root in [0,1]\left[0,1\right]?
Does the equation 9x+cos⁡ (3x)=89x+\cos\ (3x)=8 have a solution in the domain (0,1)?
Show that the equation sin⁡x=x2−x−1\sin x=x^2-x-1 has at least one root in the interval (0,3)(0,3), i.e., there exist a number c∈(0,3)c \in (0,3) such that sin⁡c=c2−c−1\sin c=c^2-c-1 . State the name of the theorem you use.
🦊 TRICKY! Use the Intermediate Value Theorem to show that the function f(x)=x3−4x+2f\left(x\right)=x^3-4x+2 crosses the x-axis at least twice in the interval [0,2].
Use Intermediate Value Theorem to conclude that x2+x−3=2\sqrt{x^2+x-3}=2 has at least one solution in the interval x∈(2,3)x\in(2,3).
Does the equation 3x2+cos⁡x=33x^2+\cos{x}=3 have a solution in the domain (0,1)(0,1)?
Does the equation 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have at least one solution in the domain (0,1)?
Determine if the polynomial x3−6xx^3-6x will attain the value of -4 in the domain [0,1].
Intermediate Value Theorem
Suppose f is a continuous function and f(0)=0 and f(1)=2. Is there a constant c∈(0,1)c\in (0,1) such that f(c)=2−c5f(c)=\sqrt[5]{2-c}? HINT: Apply the Intermediate Value Theorem on the function g(x)=[f(x)]5+xg(x)=[f(x)]^5+x.
The Intermediate Value Theorem
Show that there is at least one solution to the equation ln⁡x=esin⁡(πx2)\ln x=e^{\sin\left(\frac{\pi x}{2}\right)}.
Intermediate Value Theorem
If f(x)f\left(x\right) is a continuous function on [−1,1]\left[-1,1\right], such that f(−1)=−πf\left(-1\right)=-\pi and f(1)=πf\left(1\right)=\pi, which statement is always true?
Intermediate Value Theorem
Let f(x)f(x) be an even, continuous function and suppose that
f(−2019)=12f(-2019) = \frac{1}{2}
and that
Intermediate Value Theorem
Find an interval of length at most 1 that contains a solution to the equation
ex−2=0e^x - 2 = 0
Intermediate Value Theorem
Show that the function
p(x)=x4+x2−2p(x)=x^4+x^2-2
has a solution on the interval [0,2][0, 2] .
The Intermediate Value Theorem
Does 3x2+cos⁡x=33x^2+\cos x=3 has a root in (0, 1)?
f(x):[0,1]→[1,3]f(x):[0,1]\rightarrow[1,3] is continuous. Does f(x)f(x) intersect with y=2x+1y=2x+1 ?
Alex is thinking about tan⁡x\tan x function. He knows that tan⁡1>0\tan1>0 and tan⁡2<0\tan2<0 He concludes that tan⁡x\tan x has a root in (1, 2). Justify if he is right or wrong.
More Continuity Questions:
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Continuity
Consider the piecewise function
h(x)={Cx−17x<10C+3xx=102C−4x>10h(x) = \begin{cases} \frac{C}{x-17} & x<10 \\ C+3x & x=10 \\ 2C-4 & x>10 \end{cases}
where C is the same real number in each formula.
Continuous Functions
For what values of aa and bb is the following function continuous? Answers are given in the form (a,b)\left(a,b\right).
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuous Functions
For what values of aa and bb is the following function continuous? Answers are given in the form (a,b)(a,b).
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
Use the graph below to find
Continuity
Use the graph below to find
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Practice: Evaluating limits
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right). Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\frac{1}{\ln\left(f\left(x\right)\right)} is continuous.
f(x)f\left(x\right) is discontinuous at x=0
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuous Functions
Practice: Continuous Functions
Suppose that f(x)f\left(x\right) and g(x)g\left(x\right) are continuous functions on (−3, 2)\left(-3,\ 2\right). If g(1)=−3g\left(1\right)=-3 and lim⁡x→1(ln⁡(4+g(x))−g(x)ef(x))=1\displaystyle\lim_{x\to1}\left(\ln\left(4+g\left(x\right)\right)-\frac{g\left(x\right)}{e^{^{f\left(x\right)}}}\right)=1, determine the value of f(1)f\left(1\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Logarithmic Function & Continuity
The following is the graph of y=f(x)y=f\left(x\right). On what interval is ln⁡(f(x)−1)\ln\left(f\left(x\right)-1\right) continuous?
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1if x≠−24       if x=25       if x=−2f(x)=\left\{\begin{array}{ll} \displaystyle \frac{x^3-8}{x^2-4}+1\quad\text{if}\,x\neq -2 \\ \\ 4\quad\qquad\quad\,\,\,\,\,\,\,\text{if}\,x=2\\ \\ 5\quad\qquad\quad\,\,\,\, \,\,\,\text{if}\,x=-2 \end{array}\right.
Practice: Remove Discontinuity
Q.\textbf{Q.} Consider the function f(x)=cos⁡x−1x\displaystyle f(x)=\frac{\cos{x}-1}{x}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined, and that is continuous on R\mathbb{R}.
Continuity
Assume a, b∈Ra,\ b \in \mathbb{R} and f(x)f(x) is the function given by
f(x)={ex−1,if  x>1a,if  x=1−x2+b,if  x<1f(x)=\begin{cases} e^{x-1},& \text{if}\;x>1\\ a,&\text{if}\;x=1\\ -x^2+b,&\text{if}\;x<1 \end{cases}
Continuity
Find AA so that the following function is continuous at x=1x=1:
f(x)={2x2−x+A  if  0≤x≤1x2+2x−3x2−1   if  x>1{f(x)=\left\{\begin{array}{ll} \displaystyle 2x^2-x+A\quad\quad\,\,\text{if }\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if }\,x> 1 \end{array}\right.}
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Q.\textbf{Q.} Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Consider the function f(x)=cos⁡2x−1x2f(x)=\dfrac{\cos^2{x}-1}{x^2}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined and that is continuous on all R\mathbb{R}.
Find values of aa and bb such that the following function ff is continuous at x=0x=0 and x=1x=1:
f(x)={a(1−cos⁡2xsin⁡(2x)), if x<02x2−x+b, if 0≤x≤1x2+2x−3x2−1, if x>1f(x)=\begin{cases} a\left(\dfrac{\sqrt{1-\cos^2x}}{\sin(2x)}\right)&,\text{ if }x<0\\\\ 2x^2-x+b&,\text{ if }0\le x\le1\\\\ \dfrac{x^2+2x-3}{x^2-1}&,\text{ if }x>1 \end{cases}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={x2+10;x<1−x+7a;x≥1f(x)=\begin{cases} x^2+10;&x<1\\ -x+7a;&x\geq1 \end{cases}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={−2x2+7;x<−43x+a;x≥−4f(x)=\begin{cases} -2x^2+7;&x<-4\\ 3x+a;&x\geq-4 \end{cases}
Continuity and Piecewise Functions
What values of aa and bb make the following function continuous?
f(x)={ax2−4;x<−2ax+b;−2≤x≤3ax2+bx+10;x>3f(x)= \begin{cases} ax^2-4;&&x<-2\\\\ ax+b;&&-2\leq{}x\leq3\\\\ ax^2+bx+10;&&x>3 \end{cases}
Continuity
Determine whether the following function is continuous at x=0x = 0 using the definition of continuity. You must full justify your solution.
g(x)={x2+4−2x , x<00 , x≥0\displaystyle g(x) = \begin{cases} \frac{\sqrt{x^2 + 4} - 2}{x} & \text{ , } x < 0 \\ 0 & \text{ , } x \geq 0 \end{cases}
Continuity
Where is f(x)=cos⁡(πx2)1−x2f(x) = \frac{\cos\left(\frac{\pi x}{2} \right)}{\sqrt{1 - x^2}} continuous?
Infinite Limits and Continuity
Given the function f(x)={arctan⁡x+π/2if x≤02xif 0<x≤e1ln⁡xif x>ef(x)=\begin{cases} \arctan x+\pi/2&\text{if }x \le 0\\ \frac{2}{x}&\text{if }0<x\le e\\ \frac{1}{\ln x}&\text{if }x>e \end{cases}, which of the following statements is true about f(x)f(x)?
f(x)={a2x2+1if x<1x+4if x=1−5(a+x)if x>1f\left(x\right)= \begin{cases} a^2x^2+1&if\space x<1\\ x+4 & if \space x=1\\ -5(a+x)&if~x>1 \end{cases}
a) For what value(s) of aa does lim⁡x→1f(x)\displaystyle \lim_{x\to1}f(x) exist?
b) For what value(s) of aa is f(x)f(x) continuous at x=1x=1?
Continuity
Use the graph below to find
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Practice: Continuity of Piecewise Functions
Consider the function
f(x)={ln⁡(x2)+xif  x<−1cos⁡πx2if  −1≤x<12exif  x≥12f\left(x\right)= \begin{cases} \ln(x^2)+x & \text{if }\space x<-1\\ \cos \frac{{\pi}x}{2}&\text{if }\space -1\le x<\frac{1}{2}\\ e^{x}&\text{if }\space x\ge \frac{1}{2} \end{cases}
Limits: Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1 if x≠−2, x≠24 if x=25 if x=−2f(x) = \begin{cases} \frac{x^3-8}{x^2-4}+1 & \text{ if } x\neq -2, \, x \neq 2 \\ 4 & \text{ if } x = 2 \\ 5 & \text{ if } x = -2 \end{cases}
Limits: Continuity
Find A and B so that the following function is continuous at 0 and 1:
f(x)={A(1−cos⁡2xsin⁡2x)x<02x2−x+B0≤x≤1x2+2x−3x2−1x>1f(x) = \begin{cases} A\left(\frac{\sqrt{1-\cos^2 x}}{\sin 2x}\right) & x < 0 \\ 2x^2-x+B & 0 \leq x \leq 1 \\ \frac{x^2+2x-3}{x^2-1} & x > 1 \end{cases}
Enter your answer as A,B. For instance, if you think A=0, B=1, enter 0,1
If f(x)f(x) is continuous at x=−2x=-2 and if lim⁡x→−2(2f(x)−x)2∣f(x)∣2+1=4\lim\limits_{x\rightarrow -2}\frac{(2f(x)-x)^2}{|f(x)|^2+1}=4, then
Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1if x≠−24       if x=25       if x=−2f(x)=\left\{\begin{array}{ll} \displaystyle \frac{x^3-8}{x^2-4}+1\quad\text{if}\,x\neq -2 \\ \\ 4\quad\qquad\quad\,\,\,\,\,\,\,\text{if}\,x=2\\ \\ 5\quad\qquad\quad\,\,\,\, \,\,\,\text{if}\,x=-2 \end{array}\right.
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Q.\textbf{Q.} Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Practice: Remove Discontinuity
Q.\textbf{Q.} Consider the function f(x)=cos⁡x−1x\displaystyle f(x)=\frac{\cos{x}-1}{x}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined, and that is continuous on R\mathbb{R}.
Continuity
Assume a, b∈Ra,\ b \in \mathbb{R} and f(x)f(x) is the function given by
f(x)={ex−1,if  x>1a,if  x=1−x2+b,if  x<1f(x)=\begin{cases} e^{x-1},& \text{if}\;x>1\\ a,&\text{if}\;x=1\\ -x^2+b,&\text{if}\;x<1 \end{cases}
Continuity
Find AA so that the following function is continuous at x=1x=1:
f(x)={2x2−x+A  if  0≤x≤1x2+2x−3x2−1   if  x>1{f(x)=\left\{\begin{array}{ll} \displaystyle 2x^2-x+A\quad\quad\,\,\text{if }\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if }\,x> 1 \end{array}\right.}
Find all number(s) C that make(s) f(x) continuous.
f(x)={x−C1+Cx≤02x2+Cx>0f(x)=\begin{cases}\displaystyle\frac{x-C}{1+C}&x\leq0\\2x^2+C&x>0\end{cases}
Discuss where the function is continuous: f(x)=sin⁡(x+cos⁡x)f(x)=\sin(x+\cos x).
If f(x)f(x) is continuous at x=2x=2 and if lim⁡x→23f(x)+2xf(x)−2=4\lim\limits_{x\rightarrow 2}\frac{3f(x)+2}{xf(x)-2}=4, then
Continuity
Find aa such that f(x)f(x) is continuous at x=1x = 1
f(x)={4x2−3ln⁡xx≥13x3−4axx<1f(x)= \begin{cases} 4x^2 - 3\ln{x} & x \ge 1 \\ 3x^3 - 4ax & x < 1 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
Consider the piecewise function
h(x)={Cx−17x<10C+3xx=102C−4x>10h(x) = \begin{cases} \frac{C}{x-17} & x<10 \\ C+3x & x=10 \\ 2C-4 & x>10 \end{cases}
where C is the same real number in each formula.
Continuity
Discuss where the following function is continuous:
h(x)=log⁡(x+1−2)h(x)=\log(\sqrt{x+1}-2)
Consider f(x)=sin⁡2xx\displaystyle f(x)=\frac{\sin{2x}}{x} . Does lim⁡x→0sin⁡2xx\displaystyle \lim_{x\rightarrow 0}\frac{\sin{2x}}{x} exist? Is ff continuous at x=0x=0?
When is the following function continuous: g(x)=x2−1x2−1\displaystyle g(x)=\frac{x^2-1}{x^2-1}?
Where is the following function continuous?
f(x)=ln⁡(x2−1)f(x)=\ln(x^2-1)
Find aa,bb such that f(x)f(x) is continuous everywhere.
f(x)={x2+x+ax>1x−b0≤x≤1x3−4x+5x<0f(x)=\begin{cases}x^2+x+a&x>1\\x-b&0\leq x\leq1\\x^3-4x+5&x<0\end{cases}
🦊 TRICKY! Find A > 0 and B so that the following function is continuous at x = 0 and x = 1:
f(x)={x2tan⁡2Axifx<03x2+2x+Bif0≤x≤13x2−7x−20x2−3x−4ifx>1f(x)=\begin{cases} \frac{x^2}{\tan^2Ax}&\text{if}&x<0\\ 3x^2+2x+B&\text{if}&0\leq x\leq 1\\ \frac{3x^2-7x-20}{x^2-3x-4}&\text{if}&x>1\end{cases}
Continuity
Consider the function f(x)=cos⁡2x−1x2f(x)=\dfrac{\cos^2{x}-1}{x^2}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined and that is continuous on all R\mathbb{R}.
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Limits: Continuity of piecewise functions
Identify the type of discontinuity (or state the function is continuous) at x=5,10,15x=5, 10, 15. Is there another point of discontinuity in the domain other than at the stated x-values?
f(x)={x+3x−6x−6+7ln⁡(15−x)sin⁡xif−∞<x≤5if 5<x≤10if10<x<15if15≤x<∞f(x)=\begin{cases} x+3\\ \frac{x-6}{x-6}+7\\ \ln(15-x)\\ \sin x \end{cases}\begin{array}{l} if -\infty < x \leq 5\\ if\ 5 < x \leq 10\\ if 10 < x< 15\\ if 15 \leq x < \infty \end{array}
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Continuity
Find the value of aa such that the function
f(x)={x2+1if x≤−3ax−1if x>−3f\left(x\right)= \begin{cases} x^2+1&if\space x\le -3\\ ax-1 & if \space x>-3 \end{cases}
is continuous everywhere.
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right), which is defined only defined on [−5,9][-5,9]. Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\dfrac{1}{\ln\left(f\left(x\right)\right)} is continuous.
Practice: Evaluating limits
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right). Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\frac{1}{\ln\left(f\left(x\right)\right)} is continuous.
f(x)f\left(x\right) is discontinuous at x=0
Continuous Functions
Practice: Continuous Functions
Suppose that f(x)f\left(x\right) and g(x)g\left(x\right) are continuous functions on (−3, 2)\left(-3,\ 2\right). If g(1)=−3g\left(1\right)=-3 and lim⁡x→1(ln⁡(4+g(x))−g(x)ef(x))=1\displaystyle\lim_{x\to1}\left(\ln\left(4+g\left(x\right)\right)-\frac{g\left(x\right)}{e^{^{f\left(x\right)}}}\right)=1, determine the value of f(1)f\left(1\right).
Logarithmic Function & Continuity
The following is the graph of y=f(x)y=f\left(x\right). On what interval is ln⁡(f(x)−1)\ln\left(f\left(x\right)-1\right) continuous?
Each question is worth 3 marks.
Continuity
What value of aa makes the following function continuous?
f(x)={2x+4,x≤2ax2+x−2,2<xf(x) = \left \{ \begin{array}{ll} 2^x + 4, & x \leq 2\\ ax^2 + x - 2, & 2 < x \end{array} \right .
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
When is the following function continuous?
e3x+1sin⁡(πx)\displaystyle \frac{e^{3x}+1}{\sin(\pi x)}