Home / AP Calculus AB: 2.2 Defining the Derivative of a Function and Using Derivative Notation  – Exam Style questions with Answer- MCQ

AP Calculus AB: 2.2 Defining the Derivative of a Function and Using Derivative Notation  – Exam Style questions with Answer- MCQ

Question

Which of the following is an equation of the line tangent to the graph of y=cosx at  \(x=\frac{\pi }{2}\)?
A \(y=x+\frac{\pi }{2}\)
B\( y=x-\frac{\pi }{2}\)
C \(y=-x+\frac{\pi }{2}\)
D \(y=-x-\frac{\pi }{2}\)

Answer/Explanation

Ans: C

Question

At time t=0, a storage tank is empty and begins filling with water. For t>0 hours, the depth of the water in the tank is increasing at a rate of W(t) feet per hour. Which of the following is the best interpretation of the statement W′(2)>3 ?

A Two hours after the tank begins filling with water, the depth of the water is increasing at a rate greater than 33 feet per hour.

B Over the first two hours after the tank begins filling with water, the depth of the water is always increasing at a rate greater than 33 feet per hour.

C Two hours after the tank begins filling with water, the rate at which the depth of the water is rising is increasing at a rate greater than 33 feet per hour per hour.

D Over the first two hours after the tank begins filling with water, the rate at which the depth of the water is rising is always increasing at a rate greater than 33 feet per hour per hour.

Answer/Explanation

Ans:C

 In the expression W′(2), the 22 represents the value of the independent variable and is therefore the number of hours since the tank began filling with water. W′(2), being the value of a derivative, is the rate of change of W, that is, the rate of change of the rate at which the depth of the water is rising; in this case, 22 hours after the tank begins filling with water. The units for the derivative would be the units of WW per unit of time; thus, feet per hour per hour. The statement says that at time 22 hours after the tank begins filling with water, the rate at which the depth of the water is rising, W(t), is increasing at a rate that is greater than 33 feet per hour per hour.

Question

When x=2e, is

A \(\frac{1}{2e}\)

B 1

C ln(2e)

D nonexistent

Answer/Explanation

Ans:A

Question

Let f be the function defined by \(f(x)=\ln (x^{2}+1)\) and let g be the function defined by \(g(x)=x^{5}+x^{2}\).The line tangent to the graph of f at x = 2 is parallel to the line tangent to the graph of g at x = a, where a is a positive constant. What is the value of a ?

A 0.246

B 0.430

C 0.447

D 0.790

Answer/Explanation

Ans:C

Question

An equation of the line tangent to the graph of \(f(x)=x(1-2x)^{3}\) at the point (1,−1) is ,

A y=−7x+6

B y=−6x+5

C y=−2x

D y=2x−3

E y=7x−8

Answer/Explanation

Ans:E

Question

The figure presents the graph of a curve and a line tangent to the curve in the x y coordinate plane, with the origin labeled O. The numbers 1 through 6, in increments of 1, are indicated on both axes. The curve is labeled y equals f of x. The curve begins on the y axis at 1, and moves upward and to the right. It begins to level off, and reaches a maximum at the point with approximate coordinates 2.8 comma 4.2. The curve then moves downward and to the right, passing through the point with coordinates 3 comma 4. It continues downward until it ends on the x axis at approximately 4.8. The line is tangent to the curve at the point with coordinates 3 comma 4. The line also passes through the points with coordinates 1 comma 5 and 5 comma 3.

Shown above is the graph of the differentiable function f along with the line tangent to the graph of f at x=3. What is the value of f′(3)?

A \(-\frac{1}{2}\)

B -2

C 4

D  \(\frac{11}{2}\)

Answer/Explanation

Ans:A

The derivative of f at x=3 is the slope of the line tangent to the graph of ff at x=3. The slope of the line is \(\frac{\Delta y}{\Delta x}=\frac{4-3}{3-5}=-\frac{1}{2}\).

 

 

Question

Let f be a differentiable function with f(1)=−2. The graph of f′, the derivative of f, is shown above. Which of the following statements is true about the line tangent to the graph of f at x=1?

A The tangent line has slope −1 and passes through the point (1,−2).

B The tangent line has slope −1 and passes through the point (1,−1).

C The tangent line has slope 0 and passes through the point (1,−2).

D The tangent line has slope 0 and passes through the point (1,−1)

Answer/Explanation

.Ans:A

 The slope of the line tangent to the graph of ff at x=1is given by f′(1). From the graph, f′(1)=−1. Because f(1)=−2, the tangent line passes through the point (1,−2).

Question

Shown above is the graph of the differentiable function f, along with the line tangent to the graph of f at x=2. What is the value of f′(2) ?

A \(\frac{1}{2}\)

B 2

C 3

D 4

Answer/Explanation

Ans:A

The derivative of  f at x=2 is the slope of the line tangent to the graph of f at x=2. The slope of the line is \(\frac{\Delta y}{\Delta x}=\frac{4-3}{4-2}=\frac{1}{2}\).

 

 

Question

The graph of a function f with f(b)>f(a) is shown above for a≤x≤b.  The derivative of f exists for all x in the interval   a<x<b except x=0.For how many values of c, for a<c<b does  ?

A Zero

B One

C Two
D Three

Answer/Explanation

Ans:D

The expression \(\frac{f(b)-f(a)}{b-a}\)  is the average rate of change of f over the interval a≤x≤b and is equal to the slope of the secant line passing through the points (a,f(a)) and (b,f(b)). The expression is the instantaneous rate of change of f at x=c and is equal to the slope of the line tangent to the graph of ff at x=c. There are four points on the graph of f at which the slope of the tangent line is equal to the slope of the secant line. Two of the points occur to the left of the y-axis, and two of the points occur to the right of the y-axis.

Question

The graph of a function f with f(b)>f(a) is shown above for a≤x≤b. The derivative of ff exists for all x in the interval a<x<b except x=0.For how many values of c, for a<c<b, does   ?

A Zero

B Two

C Three

D Four

Answer/Explanation

Ans:D

The expression \(\frac{f(b)-f(a)}{b-a}\) is the average rate of change of ff on the interval a≤x≤b and is equal to the slope of the secant line passing through the points (a,f(a)) and (b,f(b)). The expression is the instantaneous rate of change of  f at x=c and is equal to the slope of the line tangent to the graph of f at x=c. There are four points on the graph of f at which the slope of the tangent line is equal to the slope of the secant line. Two of the points occur slightly to the left of the interior local maxima. Two of the points occur slightly to the right of the local minima with horizontal tangents.

 

 

 

Question

Let f be the function defined above. Which of the following statements is true?

A f is not differentiable at x=5 because f is not continuous at x=5.

B f is not differentiable at x=5  because the graph of f has a sharp corner at x=5.

C f is not differentiable at x=5 because the graph of f has a vertical tangent at x=5.

D f is not differentiable at x=5 because ff is not defined at x=5.

Answer/Explanation

Ans:D

The limits   and are not equal. Therefore, f is not continuous at x=5, and the graph of ff would not have a sharp corner at x=5.

are not equal. Therefore, f is not continuous at x=5, and the graph of f would not have a sharp corner at x=5.

 

Question

The figure above shows the graph of a function f, which has a vertical tangent at x=4 and a horizontal tangent at x=5. Which of the following statements is false?

A  f is not differentiable at x=2 because the graph of f has a jump discontinuity at x=2.

B f is not differentiable at x=4 because the graph of f has a vertical tangent at x=4.

C f is not differentiable at x=5 because the graph of f has a horizontal tangent at x=5.

D f is not differentiable at x=7 because the graph of f has a removable discontinuity at x=7.

Answer/Explanation

Ans:C

A function is differentiable at points at which the line tangent to the graph is horizontal.

 

Question

The graph of the function f shown in the figure above has a vertical tangent at the point (2,0) and horizontal tangents at the points (1, -1) and (3,1) . For what values of x, -2<x<4, is f not differentiable?

A 0 only

B 0 and 2 only

C 1 and 3 only

D 0,1 and 3 only

E 0,1,2 and 3

Answer/Explanation

 Ans:B

Question

The graph of a function f is shown above. At which value of x is f continuous, but not differentiable?

A a

B b

C c

D d

E e

Answer/Explanation

Ans:a

Question

A I only

B II only

C III only

D I and II only

E I,II and III

Answer/Explanation

Ans:A

Question

Let f be the function given by \(f(x)=(2x-1)^{5}(x+1)\) .Which of the following is an equation for the line tangent to the graph of f at the point where  x=1?

A y=21x+2

B y=21x−19

C  y=11x−9

D y=10x+2

E y=10x−8

Answer/Explanation

Ans:B

Question

If  \( f(x)=2+|x-3|\)for all x, then the value of the derivative f ′(x )  at x = 3 is.

(A) -1                             (B) 0                                   (C) 1                    (D) 2                                     (E) nonexistent

Answer/Explanation

Ans:E

Question

If \(y=tan u\) ,u=\(v-\frac{1}{v}\), and v= ln x, what is the value of x, what is the value of \(\frac{dy}{dx}at x=e\)?

(A)0                                (B)\(\frac{1}{e}\)                               (C)1                                   (D)\(\frac{2}{e} \)                                         (E)\(sec^{2}e\)

Answer/Explanation

Ans:D

Question

The slope of the line tangent to the graph of  \(y=In(x^{2})\) at \(x=e^{2}\) is
(A)\(\frac{1}{e^{2}}\)                (B)\(\frac{2}{e^{2}}\)                    (C)\(\frac{4}{e^{2}}\)                          (D)\(\frac{1}{e^{4}}\)                 (E)\(\frac{4}{e^{4}}\)

Question

The slope of the line tangent to the graph of  \(y=In(x^{2})\) at \(x=e^{2}\) is
(A)\(\frac{1}{e^{2}}\)                (B)\(\frac{2}{e^{2}}\)                    (C)\(\frac{4}{e^{2}}\)                          (D)\(\frac{1}{e^{4}}\)                 (E)\(\frac{4}{e^{4}}\)

Answer/Explanation

Ans:A

Question

let \(f(x)=\left | sinx -\frac{1}{2} \right |\).
(A)\( \frac{1}{2}\)                      (B) 1                         (C) \(\frac{3}{2}\)                          (D)\(\frac{\pi }{2}  \)                                 (E)  \(\frac{3\pi }{2}\)

Answer/Explanation

Ans:A

Question

\(\frac{dy}{dx}=4y\) and if y=4 when x=0 then y=
(A)\(4e^{4x}\)                             (B)\(e^{4x} \)                             (C)\(3+e^{4x}\)                                (D)\(4+e^{4x}\)                        (E) \(2x^{2x}+4\)

Answer/Explanation

Ans:E

Question

If \(f(x)=x, then f'(5)\)=

(A) 0                                (B) \(\frac{1}{5}\)                                    (C) 1             (D) 5                                        (E) 25

Answer/Explanation

Ans:C

Question

The slope of the line tangent to the graph of \(y=In\left ( \frac{x}{2} \right ) \)at x=4 is

(A) \(\frac{1}{8}\)                    (B)\(\frac{1}{4}\)                               (C)\(\frac{1}{2}\)                                    (D) 1                                            (E) 4

Answer/Explanation

Ans:B

Question

\(\frac{d}{dx}\left ( \frac{1}{x^{3}}-\frac{1}{x}+x^{2} \right )\)   at  x= -1 is

(A) −6                                                   (B) -4                       (C) 0                                 (D) 2                                             (E) 6

Answer/Explanation

Ans:B

Question

An equation of the line tangent to the graph of  \(f ( x)= x(1 − 2x)^{3}\) at the point (1,-1 ) is

(A)  y =−7 x+6                 (B) y=-6x+5                              (C) y=-2x+1                       (D)y=2x-3                     (E)y=7x-8

Answer/Explanation

Ans:A

Question

 If f(x ) =sinx,  then \(f’\left ( \frac{\pi }{3} \right )=\)

(A) \(-\frac{1}{2}\)                        (B) \(\frac{1}{2} \)                                    (C)\(\frac{\sqrt{2}}{2}\)                                     (D)\(\frac{\sqrt{3}}{2}\)                         (E)\(\sqrt{3}\)

Answer/Explanation

Ans:B

Question

If \(f(x)=\sqrt{2x}\), then \(f'(2)=\)

(A)\( \frac{1}{4}\)                             (B)\(\frac{1}{2}\)                 (C)\(\frac{\sqrt{2}}{2}\)                    (D) 1                    (E) \(\sqrt{2}\)

Answer/Explanation

Ans:B

Question

If \(f(x)=(x^{2}-2x-1)^{\frac{2}{3}}\), then f'(0)is
(A)\(\frac{4}{3}\)                             (B)0                         (C)\(-\frac{2}{3}\)                                     (D)\(-\frac{4}{3}\)                                           (E)-2

Answer/Explanation

Ans:A

Question

 If f is a differentiable function, then f ′( a)  is given by which of the following?

I.\(\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}\)
II.\(\lim_{h\rightarrow a}\frac{f(a+h)-f(x-a)}{h}\)
III.\(\lim_{h\rightarrow a}\frac{f(a+h)-f(x-a)}{h}\)

(A) I only                        (B) II only                             (C) I and II only                     (D) I and III only                              (E) I, II, and III

Answer/Explanation

Ans:C

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