Check a few more CALC questions, please?

I feel pretty good on my answers. Are there any I got wrong? Thank you for the help!

1. If g(x) = tan (5x^2), then g′(x) =

* 10x sec2(5x2)

2. The inflection point of the curve y = x4 − 8x3 + 24x2 + 7x − 3 is:

(−1, 23)
(0, 3)
(1, 21)
*(2, 59)
There is no point of inflection.

3. Suppose the functions f and g and their derivatives have the following values at x = 1 and x = 2. Let h(x) = f(g(x)). Evaluate h′(1).

x|f(x) g(x) f'(x) g'(x)
1| 8 2 1/3 -3
2| 3 -4 2pi 5

* -15

4. The slope of the tangent line to the graph y=(x^3/3)-x at the point (1,-2/3)is:

* -2/3

5. The derivative of y=cosx/1+sinx is:


tan x



* sinx/-cosx

6. Which of the following functions is continuous but not differentiable at x = 1?
I. y=^3(sqrt x-1)

II. y={x^2, x ≤ 1
{2x, x > 1

III. y={x, x ≤ 1
{1/2, x > 1
I only
II only
* I and III only
II and III only
All of these functions are continuous but not differentiable at x = 1.

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  1. #1. Since d/dx tan x = sec^2 x, you have
    10x sec^2(5x^2)

    #2. ok

    #3. ok

    #4. y' = x^2-1
    so, 0

    y = cos(1+sin)
    y' = [(-sin)(1+sin)-cos(cos)]/(1+sin)^2
    = (-sin-sin^2-cos^2)/(1+sin)^2
    = (-sin-1)/(1+sin)^2
    = -1/(1+sin)

    I. No idea what ^3(sqrt x-1) means
    II. y(1) = 1 or 2, so not continuous
    III. y(1) = 1 or 1/2, so not continuous

    So, only possible choice is I or none.

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