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Intervals of Increase and Decrease

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Wize University Calculus 1 Textbook > Applications of Differentiation

Intervals of Increase and Decrease

3 Activities

Wize University Calculus 1 Textbook > Integrals

The Fundamental Theorem of Calculus - Part 1

5 Activities

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
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.
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: 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
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.
More The Fundamental Theorem of Calculus - Part 1 Questions:
Practice: FTC I (~F2016 Final Q40)
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Derivative of Integral
Find ddx∫x3(t2−tan⁡t)100dt\displaystyle\frac{d}{dx}\int_x^3\left(t^2-\tan t\right)^{100}dt
FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Practice: FTC I
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Application of Fundamental Theorem of Calculus
For what values of xx is the function f(x)=∫x22(t−1)dtf\left(x\right)=\int_{x^2}^2\left(t-1\right)dt decreasing?
Practice: Application of FTC I (~F2015 Final Q16)
Practice: Application of FTC I
Given that lim⁡x→0∫0x(t2−1)dt=0, \lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\ determine lim⁡x→0∫0x(t2−1)dtsin⁡x\lim_{x\rightarrow0}\frac{\int_0^x\left(t^2-1\right)dt}{\sin x}.
Practice: FTC I (~F2016 Final Q40)
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Practice: FTC I
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
simona integral ftc
If g(x)=∫1xcos⁡t dt\displaystyle g\left(x\right)=\int_1^x\cos t \ dt, find g′(π2)g'\left(\dfrac{\pi}{2}\right).
Evaluate Derivative of Integral
Given f(x)=ddt∫sin⁡xcos⁡xln⁡t dt\displaystyle f(x)=\frac{d}{dt}\int_{\sin x}^{\cos x}\ln t\ dt , find f(π/3)f(\pi/3).
Practice: Deriv of Integral with Chain Rule
Q.\textbf{Q.} Find the derivative of ∫0ln⁡(x2)sin⁡t2 dt\displaystyle \int_{0}^{\ln(x^2)}\sin{t^2}\,\text{d}t.
Derivative of Integral
Find ddx∫x3(t2−tan⁡t)100dt\displaystyle\frac{d}{dx}\int_x^3\left(t^2-\tan t\right)^{100}dt
Practice: Application of FTC I (~F2018 Final Q40)
Practice: Application of FTC I
For what values of xx is the function f(x)=∫x22(t−1)dtf\left(x\right)=\int_{x^2}^2\left(t-1\right)dt decreasing?
Applications of Integration: Position, Velocity and Acceleration
If the displacement of a particle is given by s(t)=∫0tx2sin⁡xdxs\left(t\right)=\int_0^t\frac{x^2}{\sin x}dx, then the acceleration at at t=π2t=\frac{\pi}{2} is
Find ddx(∫x3x20 dt)\displaystyle\frac{d}{dx}\left(\int_x^{3x^2}0\ dt\right).
Evaluate lim⁡x→0∫0xcos⁡2t dt2x\displaystyle\lim_{x\rightarrow0}\frac{\int_0^x\cos^2t\ dt}{2x} if it is known that lim⁡x→0∫0xcos⁡2t dt=0\displaystyle\lim_{x\rightarrow0}\int_0^x\cos^2t\ dt=0.
Practice: FTC
Find the derivative of ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.
Practice: FTC
Q.\textbf{Q.} Find the derivative of ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Find the derivative using the Fundamental Theorem of Calculus.
ddx∫x3csc⁡t dt\displaystyle\frac{d}{dx}\int_x^3\csc t\ dt
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Derivatives
Find the derivative of
∫−12x3cos⁡t2 dt\int_{-1}^{2x^3} \cos t^2\ dt
Derivatives
Find the derivative ddx∫0sin⁡πsin⁡t2 dt\displaystyle \dfrac{d}{dx}\int_{0}^{\sin\pi}\sin{t^2}\,\text{d}t.
Derivatives
Given that lim⁡x→0∫0x(t2−1)dt=0, \lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\ determine lim⁡x→0∫0x(t2−1)dtsin⁡x\lim_{x\rightarrow0}\frac{\int_0^x\left(t^2-1\right)dt}{\sin x}.
Practice: FTC I
If g(x)=∫x2excos⁡t dt\displaystyle g\left(x\right)=\int_{x^2}^{e^x}\cos t\ dt, find g′(0)g'\left(0\right).
Practice: Application of FTC I
Given that lim⁡x→0∫0x(t2−1)dt=0, \displaystyle \lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\ determine lim⁡x→0∫0x(t2−1)dtsin⁡x\displaystyle \lim_{x\rightarrow0}\frac{\displaystyle \int_0^x\left(t^2-1\right)dt}{\sin x}.
Find the derivative using the Fundamental Theorem of Calculus: ddx∫sin⁡xcos⁡xln⁡t dt\displaystyle\frac{d}{dx}\int_{\sin x}^{\cos x}\ln t\ dt.
Find the derivative using the Fundamental Theorem of Calculus: ddx∫x3csc⁡t dt\displaystyle\frac{d}{dx}\int_x^3\csc t\ dt.
Compute the derivative ddx[∫x21(et+1)dt]\displaystyle \frac{d}{dx}\left[\int_{x^2}^{1}(e^t+1)dt\right].
Trigonometric Functions
Let f(x)=1+∫55xarctan⁡(t2)dtf\left(x\right)=1+\int_5^{5x}\arctan\left(t^2\right)dt
Find the linearization of f(x)f\left(x\right) centered at a=1a=1.
simona integral ftc
If g(x)=∫1xcos⁡t dt\displaystyle g\left(x\right)=\int_1^x\cos t \ dt, find g′(π2)g'\left(\dfrac{\pi}{2}\right).
Practice: Deriv of Integral with Chain Rule
Q.\textbf{Q.} Find the derivative of ∫0ln⁡(x2)sin⁡t2 dt\displaystyle \int_{0}^{\ln(x^2)}\sin{t^2}\,\text{d}t.
Evaluate Derivative of Integral
Given f(x)=ddt∫sin⁡xcos⁡xln⁡t dt\displaystyle f(x)=\frac{d}{dt}\int_{\sin x}^{\cos x}\ln t\ dt , find f(π/3)f(\pi/3).
Derivative of Integral
Find ddx∫x3(t2−tan⁡t)100dt\displaystyle\frac{d}{dx}\int_x^3\left(t^2-\tan t\right)^{100}dt
Find the derivative using the Fundamental Theorem of Calculus: ddx∫x3xarctan⁡ t dt\displaystyle\frac{d}{dx}\int_{\sqrt{x}}^{3x}\arctan\ t\ dt.
Find the derivative using the Fundamental Theorem of Calculus: ddx∫3x9xlog⁡3t dt\displaystyle\frac{d}{dx}\int_{3^x}^{9^x}\log_3t\ dt.
Find the second derivative with respect to x of ∫46xt1+tdt\int_4^{6x}\frac{t}{1+t}dt
Find the derivative of the integral, i.e.
ddx∫sin⁡xcos⁡xln⁡t dt\displaystyle\frac{d}{dx}\int_{\sin x}^{\cos x}\ln t\ dt
Find the derivative with respect to xx of ∫π/6ln⁡x(sec⁡t+tan⁡t)dt\int_{\pi/6}^{\ln x}(\sec t+\tan t)dt.
Find the second derivative with respect to xx of ∫46xt1+tdt\int_4^{6x}\frac{t}{1+t}dt.
Find ddx(∫x3x20 dt)\displaystyle\frac{d}{dx}\left(\int_x^{3x^2}0\ dt\right).
Evaluate lim⁡x→0∫0xcos⁡2t dt2x\displaystyle\lim_{x\rightarrow0}\frac{\int_0^x\cos^2t\ dt}{2x} if it is known that lim⁡x→0∫0xcos⁡2t dt=0\displaystyle\lim_{x\rightarrow0}\int_0^x\cos^2t\ dt=0.
Applications of Integration: Position, Velocity and Acceleration
A particle's velocity in cm/s is given by the function v(t)=∫1t2x dxv\left(t\right)=\int_1^{t^2}x\ dx. When is the particle's acceleration 5454cm/s2?
Derivatives
Find ddx[∫x3x(t−t3) dt]\frac{d}{dx}\left[\int_{x^3}^{\sqrt{x}}\left(\sqrt{t}-t^3\right)\ dt\right].
Application of Fundamental Theorem of Calculus
For what values of xx is the function f(x)=∫x22(t−1)dtf\left(x\right)=\int_{x^2}^2\left(t-1\right)dt decreasing?
Practice: Application of FTC I (~F2015 Final Q16)
Practice: Application of FTC I
Given that lim⁡x→0∫0x(t2−1)dt=0, \lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\ determine lim⁡x→0∫0x(t2−1)dtsin⁡x\lim_{x\rightarrow0}\frac{\int_0^x\left(t^2-1\right)dt}{\sin x}.
Practice: FTC I (~F2016 Final Q40)
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Practice: FTC
Q.\textbf{Q.} Find the derivative of ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.