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