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Application of Fundamental Theorem of Calculus

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Wize University Calculus 2 Textbook > Integration

Fundamental Theorem of Calculus (FTC)

7 Activities

Wize University Calculus 1 Textbook > Integrals

The Fundamental Theorem of Calculus - Part 1

5 Activities

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?
More Fundamental Theorem of Calculus (FTC) Questions:
Fundamental theorem of calculus
Find the derivative ddx\displaystyle \frac{d}{dx} of the integral ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.
Fundamental theorem of calculus
Find the derivative ddx\displaystyle \frac{d}{dx} of the integral ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.
Fundamental theorem of calculus
Find the derivative ddx\displaystyle \frac{d}{dx} of the integral ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.
Fundamental theorem of calculus
If ∫02[3(kx)2+(k+1)x+1]dx=10\displaystyle \int_0^2[3(kx)^2+(k+1)x+1]dx=10, then what is a possible value of kk ?
Fundamental theorem of calculus
Find the derivative ddx\displaystyle \frac{d}{dx} of the integral ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.
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?
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
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].
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.