# 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

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

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

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

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)