A) The velocity of a particle moving along a line is t^3 ? t meters per second.

Question

A) The velocity of a particle moving along a line is t^3 ? t meters per second. Find the distance traveled in meters during the time interval 1 _< t_< 2. B) If Integral (3,0) f(x) dx = 4,and Integral (6,3) f(x) dx=4, and Integral (6,2) f(x) dx=5, find the value of Integral (2,0) f(x)dx? C) Compute the derivative d/dx Integral (x,4) sin(t^3)dt using the Fundamental Theorem of Calculus part 1. D) Compute the derivative d/dx Integral (e^x^2,2) ln(t) dt using the Fundamental Theorem of Calculus part 1.

Explanation / Answer

A)

velocity=t^3-t

dx/dt=t^3-t

=>x=t^4/4-t^2/2

hence distance travelled between t=1 and t=2 is

x(t=2)-x(t=1)

=>distance=2.5m

B)

Integral (2,0) f(x)dx

=Integral (3,0) f(x) dx + Integral (6,3) f(x) dx - Integral (6,2) f(x) dx

= 8-5

=3

C)

d/dx integral (x,4) sin(t^3)dt

= sin (x^3)

D)

ln(e^x^2)* (2xe^x^2)

2x^3e^x^2

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