Home / AP Calculus AB: 5.9 Connecting a Function, Its First Derivative, and  Its Second Derivative – Exam Style questions with Answer- MCQ

AP Calculus AB: 5.9 Connecting a Function, Its First Derivative, and  Its Second Derivative – Exam Style questions with Answer- MCQ

Question

The function \(f\) is twice differentiable with \(f(2)=1, f^{\prime}(2)=4\), and \(f^{”}(2)=3\). What is the value of the approximation of \(f(1.9)\) using the line tangent to the graph of \(f\) at \(x=2\) ?

A 0.4

B 0.6

C 0.7

D 1.3

E 1.4

▶️Answer/Explanation

Ans:B

\[
f(2)=1, f^{\prime}(2)=4 f^{\prime \prime}(2)=3 x=2 f(1.9)=?
\]

The formula for the slope of a tangent line to a curve \(y=f(x)\) at the point \((a, f(a))\) is:
\[
\begin{aligned}
m & =\frac{y-f(a)}{x-a} \\
f^{\prime}(a) & =\frac{y-f(a)}{x-a}
\end{aligned}
\]

Rewrite the above expression for the variable \(y\) or the function \(f(x)\).
\[
\begin{aligned}
(x-a) f^{\prime}(a) & =y-f(a) \\
y-f(a) & =(x-a) f^{\prime}(a) \\
y & =(x-a) f^{\prime}(a)+f(a) \\
f(x) & =(x-a) f^{\prime}(a)+f(a) \quad[\because y=f(x)]
\end{aligned}
\]

Substitute \(a=2\) in the above function.
\[
\begin{array}{rlr}
f(x) & =(x-2) f^{\prime}(2)+f(2) & \\
& =(x-2) 4+1 \quad\left[\because f^{\prime}(2)=4 \text { and } f(2)=1\right] \\
& =4 x-8+1 \\
& =4 x-7
\end{array}
\]

Substitute \(x=1.9\) in the above function and solve for
\[
\begin{aligned}
& f(1.9) . \\
& \begin{aligned}
f(1.9) & =4(1.9)-7 \\
& =7.6-7 \\
& =0.6
\end{aligned}
\end{aligned}
\]

Question

The table above gives values of the continuous function f at selected values of x. If f has exactly two critical points on the open interval (10, 14) , which of the following must be true?

A f(x) > 0 for all x in the open interval (10, 14)

B f(x) exists for all x in the open interval (10, 14)

C f(x) < 0 for all x in the open interval (10, 11)

D f(12)  0

▶️Answer/Explanation

Ans:D

Question

The graph of f”,the second derivative of the function f, is shown above. Which of the following could be the graph of f ?

A

B

C

D

▶️Answer/Explanation

Ans:D

Question

The graph of f’, the derivative of the function f, is shown in the figure above. Which of the following statements must be true?

I. f is continuous on the open interval (a, b).

II f is decreasing on the open interval (a, b).

III The graph of f is concave down on the open interval (a, b).

A I only

B I and II only

C I and III only

D II and III only

▶️Answer/Explanation

Ans:C

Question

The graph of a differentiable function f is shown in the figure above. Which of the following is true?

A f(2)<f(0)<f(3)

B f(2)<f(3)<f(0)

C f(3)<f(2)<f(0)

D f(3)<f(0)<f(2)

▶️Answer/Explanation

Ans:D

Scroll to Top