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Intervals of Increase and Decrease: Rate of Change

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Wize University Calculus 1 Textbook > Derivatives

Derivatives as a Rate of Change

4 Activities

Wize University Calculus 1 Textbook > Applications of Differentiation

Intervals of Increase and Decrease

3 Activities

f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.


More Derivatives as a Rate of Change Questions:
Derivatives and Tangent Lines
A certain particle is traveling through space and time. We are able to measure the position of the particle at certain time values. Although we can't measure the particle continuously, we know that the particles position function is differentiable. The table gives values of time for the differentiable function s(t) for 0≤t≤40\le t\le4. s(t) represents the position of the particle at time t.
Practice: Derivatives as a Rate of Change
f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.
Average Rate of Change & the Secant Line
Practice: Average Rate of Change & the Secant Line
The population of Vancouver in 2000 was 450,000. In 2020, the population was 740,000. What is the average annual rate of change of the population of Vancouver?
Average Rate of Change & the Secant Line
Practice: Average Rate of Change & the Secant Line
Sherry bought a car in 2010 for $60,000\$60,000. 5050 years later, she sold the car as a collector car for $98,000\$98,000. What is the average annual rate of change of the value of the car? (Ignore units)
Average Rate of Change & the Secant Line
Practice: Average Rate of Change & the Secant Line
An investment, PP, in $, grows with interest over time tt (in years) is given by the table of values below.
t01234P10001305152817741936\begin{array}{|r|c|c|c|c|c|}\hline \textbf{t}&0&1&2&3&4\\\hline \textbf{P}&1000&1305&1528&1774&1936\\\hline \end{array}
Average Rate of Change & the Secant Line
Practice: Average Rate of Change & the Secant Line
The amount of bacteria, AA, in mg, of a certain species over time tt (in minutes) is given by the table of values below.
t01234A150183204247283\begin{array}{|r|c|c|c|c|c|}\hline \textbf{t}&0&1&2&3&4\\\hline \textbf{A}&150&183&204&247&283\\\hline \end{array}
Average Rate of Change & the Secant Line
Practice: Average Rate of Change & the Secant Line
Determine the average rate of the change of the function g(t)=−3t2t+5g(t)=\dfrac{-3t}{2t+5} over the interval 0≤t≤50\leq{}t\leq{}5.
Average Rate of Change & the Secant Line
Practice: Average Rate of Change & the Secant Line
Determine the average rate of the change of the function g(t)=t2t2−6g(t)=\dfrac{t^2}{t^2-6} over the interval 0.5≤t≤20.5\leq{}t\leq{}2.
The average rate of change of the function f(x)=2cos⁡(x)f(x)=2\cos(x) over the interval [π6,π2][ \frac{\pi}{6},\frac{\pi}{2}] is
Derivatives: Rate of Change
In recent years, exports from Canada to the United States are on the rise. Let E(t)E(t) represent the amount of exports to the United States from Canada (in Billions of dollars) at time tt. Let tt be the number of years after 2000. Interpret the following statements without using the words "derivative" or "instantaneous rate of change".
E(2)=5E(2)=5
E′(3)=1E'(3)=1
Derivatives and Tangent Lines
A certain particle is traveling through space and time. We are able to measure the position of the particle at certain time values. Although we can't measure the particle continuously, we know that the particles position function is differentiable. The table gives values of time for the differentiable function s(t) for 0≤t≤40\le t\le4. s(t) represents the position of the particle at time t.
The average rate of change of the function f(x)=3sin(x)−1f(x)=3sin(x)-1 over the interval [0,π6][0, \frac{\pi}{6}] is
More Intervals of Increase and Decrease Questions:
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=−1x+1\displaystyle f(x)=-\frac{1}{x+1} is increasing or decreasing.
Intervals of Increase and Decrease
Find the intervals in which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
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.
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Q:\bf{Q:} Which of the following functions is/are always increasing?
Practice: Derivatives as a Rate of Change
f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.
Intervals of Increase and Decrease
If the function is given by f(x)=∫x2(t+1)2(t−2)t dtf(x)=\int_x^2\left(t+1\right)^2\left(t-2\right)t\ dt, then f(x)f\left(x\right) is decreasing on the interval
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?
Intervals of Increase and Decrease
f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.
Concavity and Inflection Points
Find the intervals of increase/decrease and the intervals of concavity for f(x)=ln⁡(x2−1)f(x)=\ln(x^2-1)
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
Increasing Interval: Relative extrema
If the derivative of the function ff is f′(x)=ex(x−3)2(x+1)(2−x)f'\left(x\right)=e^x\left(x-3\right)^2\left(x+1\right)\left(2-x\right), then f(x)f\left(x\right) is increasing on the interval?
Intervals of Increase and Decrease
Find the intervals in which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
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.
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=−1x+1\displaystyle f(x)=-\frac{1}{x+1} is increasing or decreasing.
Q:\bf{Q:} Which of the following functions is/are always increasing?
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}
f′(x)f'(x) plot for a differentiable function is depicted below.
Find interval(s) at which ff is increasing.
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.
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}
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
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)
Exercises
Find the intervals in which the function f(x)=−1x+1\displaystyle f(x)=\frac{-1}{x+1} is increasing or decreasing.
f′(x)f'(x) plot for a differentiable function is depicted as below. Answer the following questions.
Find interval(s) at which ffis increasing.