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Applications of Integration: Position, Velocity and Acceleration

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Wize University Calculus 1 Textbook > Integrals

The Fundamental Theorem of Calculus - Part 1

5 Activities

Wize University Calculus 1 Textbook > Applications of Differentiation for Science

Position, Velocity, and Acceleration

3 Activities

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
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
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.
More Position, Velocity, and Acceleration Questions:
Applications of Integration: Position, Velocity and Acceleration
If the accelertion of a particle is given by a(x)=2x+xa\left(x\right)=2^x+x, and v(0)=s(0)=0v\left(0\right)=s\left(0\right)=0, then the displacement of the particle s(x)s\left(x\right) is
Practice: Velocity and Acceleration
A point PP moves along the x-axis. Its position after tt seconds is given by x=2t3−21t2+60tx=2t^3-21t^2+60t meters.
Practice: Velocity and Acceleration
Q.\textbf{Q.} A point PP moves along the x-axis. Its position after tt seconds is given by x=2t3−21t2+60tx=2t^3-21t^2+60t meters.
Practice: Velocity and Acceleration
Q.\textbf{Q.} A point PP moves along the x-axis. Its position after tt seconds is given by x=2t3−21t2+60tx=2t^3-21t^2+60t meters.
Applications of derivatives: Velocity and Direction
At time t≥0t\ge0, the velocity (in m/s) of a particle moving along a horizontal line is v(t)=t2−2t+2v(t)=t^2-2t+2
Mean Value Theorem: Speed
In 2 hours, a car driver covered 150 km on a road with speed limit 65 km per hour. Did the car go over the speed limit? Answer using the mean-value theorem.
Applications of derivatives: Velocity and Direction
At time t≥0t\ge0, the velocity (in m/s) of a particle moving along a horizontal line is v(t)=t2−2t+2v(t)=t^2-2t+2
Written Answer 3
The velocity of a protein inside a cell starting at the top and falling under the force of gravity is given by the differential equation
dvdt=g−kv\displaystyle \frac{dv}{dt} = g - kv
where g≈107μm/s2g \approx 10^7 \mu m / s^2 is the gravitational acceleration. kvkv is the acceleration due to drag and k=1013s−1k = 10^{13} s^{-1}. (Note that this model ignores diffusion of the protein which makes it a really bad model but we're exploring what gravity alone would do to a protein.
Applications of Integration: Position, Velocity and Acceleration
If the acceleartion of a particle is given by a(x)=2x+xa\left(x\right)=2^x+x, and v(0)=s(0)=0v\left(0\right)=s\left(0\right)=0, then the displacement of the particle s(x)s\left(x\right) is
Applications of Integration: Position, Velocity and Acceleration
The velocity of a particle is given by v(t)=∫05f(x)dxv\left(t\right)=\int_0^5f\left(x\right)dx, the acceleration of this particle is
Related Rates
An object is moving in such a way that its position is given by x(t)=2t3−tx(t)=2t^3-t Find the time ttat which the object’s speed is equal to the average speed over the interval [0, 1].
Kinematics: Free fall
A ball at rest is dropped off a 150-m tall building. Ignoring air resistance, what will its speed be the instant it hits the ground?
Enter your answer as a positive value (speed) in m/s, do not include units.
Mean Value Theorem: Speed
In 2 hours, a car driver covered 150 km on a road with speed limit 65 km per hour. Did the car go over the speed limit? Answer using the mean-value theorem.
Practice: Velocity and Acceleration
Q.\textbf{Q.} A point PP moves along the x-axis. Its position after tt seconds is given by x=2t3−21t2+60tx=2t^3-21t^2+60t meters.
Applications of derivatives: Velocity and Direction
At time t≥0t\ge0, the velocity (in m/s) of a particle moving along a horizontal line is v(t)=t2−2t+2v(t)=t^2-2t+2
An object is moving according to x(t)=t4−8t2+2x(t)=t^4-8t^2+2. Find the time at which its acceleration is zero.
x(t)=6t3+3t2−3,x(t)=6t^3+3t^2-3, Find velocity and acceleration at t=2?
Applications of Integration
A particle moves in a straight line so that its velocity at time tt is v(t)=t+1v(t)=\sqrt{t+1}. If its position at t=8t=8 is s(8)=20s(8)=20, find s(9)s(9).