AP Calc BC Differentiation Fundamentals

Question 1

Let f(x) = x^3. Which of the following limits equals f'(2)?

Accepted Answer

lim(h→0) [(2+h)^3 − 2^3] / h, which equals 12

Answer

Expanding (2+h)^3 − 8 gives 12h + 6h^2 + h^3. Dividing by h and letting h→0 gives 12, which matches f'(2) = 3(2)^2 = 12.

Question 2

If y = ln(cos(3x)), what is dy/dx?

Accepted Answer

−3 tan(3x)

Answer

Apply the chain rule twice: d/dx ln(u) = u'/u with u = cos(3x), and u' = −3 sin(3x). This gives −3 sin(3x)/cos(3x) = −3 tan(3x).

Question 3

The curve x^2 + xy + y^2 = 7 passes through the point (1, 2). What is dy/dx at that point?

Accepted Answer

−4/5

Answer

Differentiating implicitly gives 2x + y + x(dy/dx) + 2y(dy/dx) = 0. At (1, 2) this becomes 4 + 5(dy/dx) = 0, so dy/dx = −4/5.

Question 4

Let f(x) = x^3 + x + 1, which is one-to-one. If g is the inverse function of f, what is g'(3)?

Accepted Answer

1/4

Answer

Since f(1) = 3, g(3) = 1. Then g'(3) = 1/f'(g(3)) = 1/f'(1) = 1/(3·1 + 1) = 1/4.

Question 5

What is the value of the derivative of f(x) = arctan(2x) at x = 1/2?

Accepted Answer

1

Answer

d/dx arctan(2x) = 2/(1 + (2x)^2). At x = 1/2 this is 2/(1 + 1) = 1.

AP Calc BC Practice Test