## Advanced Placement Calculus Answers

- Question #1: The function given by:

is differentiable at x=0. What is the value of f(n-m)?

(a) 2+e

(b) 3+e

^{2}(c) e

^{2}(d) 2e

(e) e

^{3}Solution: Since

*f(x)*is differentiable at*x = 1*, it must be continuous there, which means thatf(0) = n+1

n + 1 = 4, and n = 3

Now we can determine

*m*from the fact that*f(x)*is differentiable at*x = 0*.The right-hand branch slope is

and f'(0) = 2.

The left-hand branch slope is f'(0) = m.m = 2

At this point we can calculate n – m = 1

f(1) = 3+e

^{2} - Question #2: If y = sin(x
^{3}), what is dy/dx?

(a) 3x^{2}cos(x^{3})(b) -3x

^{2}cos(x^{3})(c) x

^{2}cos(3^{2})(d) -x

^{2}cos(3^{2})(e) cos(x

^{3})Answer:

u(x) = x^{3}du/dx = 3x

^{2}y(u) = sin(u)

dy/du = cos(u)

dy/dx = (dy/du)(du/dx) = cos(x

^{3})·3x^{2}dy/dx = 3x

^{2}cos(x^{3}) - Question #3:

(a) -1

(b) 1/5

(c) 1

(d) -1/3

(e) 1/3

Answer:

- Question #4: What is the value of a, if:

(a) ¶

(b) 1

(c) 1 + ¶

(d) √2

(e) 1 + e

Answer: u = 2 + sin(ax)

du/dx = a·cos(ax)

ln(3/2) = (1/a)ln(3/2) and a = 1.

- Question #5: Which of the following are antiderivatives of f(x) = 2
^{x}?(a) 2

^{x}/ln(2) + ln(2)(b) 2

^{2x}/ln(2) + ln(2)(c) x

^{2}/ln(2) + 1/ln(2)(d) x + 2ln(2)

(e) 2

^{x}+ ln(2)Answer:

The easiest way to solve this problem is to calculate the derivative of all 5 possible answers. We find that answer (a) is correct:

d(2

^{x}/ln(2) + ln(2))/dx = ln(2)2^{x}/ln(2) + 0 =2^{x}