1) Find the value of the derivative of each function at the specified point. Sho

ID: 2891203 • Letter: 1

Question

1) Find the value of the derivative of each function at the specified point. Show all steps (hint: you’ll need the Chain Rule).

a. f(x) = 3x+sinx (the whole thing is being taken by the square root) at x = pi / 2. (round your answer to two decimal places).

b. g(x) = 7e^x2 at x = 0.5 (round your answer to two decimal places)

2) Use the quotient rule and any trigonometric identities along the derivative rules for sine and cosine to prove that d/dx [cotx] = -csc^2x

3) At time t (where t is measured in seconds), the position of a body moving along the x-axis is given by s(t) = t^3 6t^2 + 9t meters.

a. What is the velocity of the body at t = 2 seconds? Is the body moving to the left or right at that point?

b. What is the body's acceleration at t = 2 seconds?

c. Does the particle change direction at any time during its first 10 seconds of movement? How do you know?

1. a . f(x) = sqrt(3x + sinx) = (3x + sinx)1/2

f'(x) = (1/2)(3x+ sinx)1/2 -1(3+cos x) = (3+cos x)/(2sqrt(3x+sin x))

at x= pi/2

f'(x)= (3+0)/2sqrt(3pi/2 +1)) = 3/ 2sqrt((3pi + 2)/2)

b. g(x)=7ex^2

g'(x) = 2x* 7ex^2= 14x ex^2

At x=0.5

g'(x) = 14*0.5 e0.5^2= 7e0.25

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