AP Calculus BC : 3.6 Calculating Higher- Order Derivatives- Exam Style questions with Answer- MCQ

Question

If \(y=3e^{−2x}\), then \(\frac{d^3y}{dx^3}=\)

A \(−24e^{−2x}\)

B \(−6e^{−2x}\)

C \(48e^{−2x}\)

D \(−216e^{−6x}\)

Answer/Explanation

Ans:A

This notation represents the third derivative of y . Repeated differentiation produces the following.

\(\frac{dy}{dx}=−6e^{−2x}\)

\(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})=\frac{d}{dx}(−6e^{−2x})=12e^{−2x}\)

 

Question

                               

The figure above shows the graph of f′, the derivative of the function f. At which of the four indicated values of x is f′′(x) least?
A \(A\)

B \(B\)

C \(C\)

D \(D\)

Answer/Explanation

Ans:B

 

Question

Let y=f(x) be a twice-differentiable function such that \(f(−1)=5\) and \(\frac{dy}{dx}=\frac{1}{5}(xy^2+4y)^2\). What is the value of \(\frac{d^2y}{dx^2}\) at \(x=−1\) ?
A \(−190\)

B \(−70\)

C \(−2\)

D \(10\)

Answer/Explanation

Ans:D

The second derivative can be found by using implicit differentiation of the first derivative and then evaluating at the point (−1,5).

Question

If y = sin x  and \(y^{(n)} \)means “the nth derivative of y with respect to x,” then the smallest positive  integer n for which \(y^{(n)}\)= y is

(A) 2                                         (B) 4                          (C) 5                              (D) 6                                                (E) 8

Answer/Explanation

Scroll to Top