Free Printable APPLICATION OF DERIVATIVES

1.

The curve y = x^1/5 has at (0, 0)

a)

(a) a vertical tangent (parallel to y-axis)

b)

(b) a horizontal tangent (parallel to x-axis)

c)

(c) an oblique tangent

d)

d) no tangent

2.

The equation of normal to the curve 3x² – y² = 8

which is parallel to the line x + 3y = 8 is

a)

(a) 3x – y = 8

b)

(b) 3x + y + 8 = 0

c)

(c) x + 3y ± 8 = 0

d)

(d) x + 3y = 0

3.

. If the curve ay + x² = 7 and x³ = y, cut orthogonally

at (1, 1), then the value of a is :

a)

1

b)

6

c)

0

d)

-6

4.

The tangent to the curve y = e^2x at the point (0, 1)

meets x-axis at :

a)

(0,1)

b)

(1,0)

c)

(-1/2,0)

d)

(1/2,0)

5.

$If\ f^{\ '}\left(x\right)\ \ge0\ \ \ \ ,\ \forall\ x\in R\ then\ f\ is\$

a)

strictly increasing on R

b)

strictly decreasing on R

c)

increasing on R

d)

decreasing on R

6.

$The\ function\ f\left(x\right)=e^x\ is$

a)

strictly increasing on R

b)

strictly decreasing on R

c)

increasing on R

d)

decreasing on R

7.

$The\ function\ f\left(x\right)=\ x^2-8x+12\ is\ strictly\ decrea\sin g\ in$

a)

$\left(-\infty,\ \infty\right)$

b)

$\left(2,\ 6\right)$

c)

$\left(-\infty,\ 4\right)$

d)

$\left(4,\ \infty\right)$

8.

$The\ slope\ of\ the\ \tan gent\ to\ the\ curve\ y=\ x^2+2\ at\ the\ point\ \left(2,\ 6\right)$

a)

4

b)

6

c)

-6

d)

-4

9.

$The\ \min.\ value\ of\ f\left(x\right)=x^2-6x+5\ is$

a)

4

b)

-6

c)

6

d)

-4

10.

$The\ angle\ between\ the\ lines\ y\ =\ \sqrt{3}x\ +100\ and\ \ \sqrt{3}y\ =x\ +200\ is$

a)

$60^0$

b)

$30^0$

c)

$90^0$

d)

$45^0$

11.

$The\ equation\ \ of\ the\ \ \tan gent\ \ \ to\ the\ curve\ y^2=4ax\ at\ the\ point\ (at^2,\ 2at)\ is$

a)

$ty=x+at^2$

b)

$tx+y=at^3$

c)

$ty=x–at^2$

d)

none of these

12.

The maximum and minimum value of 3 sin x + 4 cos x +10 is

a)

17 and 3

b)

15 and 5

c)

12 and 10

d)

None of these

13.

Let f (x) = x³ – 6x² + 9x + 18, then f (x) is strictly decreasing in

a)

(–∞, 1]

b)

[3, ∞)

c)

(–∞, 1] U [3, ∞)

d)

[1, 3]

14.

The absolute minimum value of x⁴ – x² – 2x+ 5

a)

is equal to 5

b)

is equal to 3

c)

is equal to 7

d)

does not exist

15.

The absolute minimum value of x⁴ – x² – 2x+ 5

a)

is equal to 5

b)

is equal to 3

c)

is equal to 7

d)

does not exist

16.

The function

2\tan^3x-3\tan^2x+12\tan x+3

,

x\ \epsilon\ \left(0,\frac{\pi}{2}\right)

is

a)

increasing

b)

decreasing

c)

increasing in $\left(0,\frac{\pi}{4}\right)$ and decreasing in $\left(\frac{\pi}{4},\frac{\pi}{2}\right)$

d)

none of these

17.

The equation of the tangent to the curve f(x) = 1 + e^-2x where it cuts the line y = 2 is

a)

2x + y = 2

b)

x - 2y = -2

c)

x + 2y = 2

d)

x - 2y = 1

18.

The maximum value of $\frac{\sin x\cos x}{\sin x+\cos x}$ in the interval $\left[0,\frac{\pi}{2}\right]$ is

a)

$\frac{1}{4}$

b)

$\frac{1}{2}$

c)

$\frac{1}{3}$

d)

$\frac{1}{2\sqrt{2}}$

19.

The minimum value of ax + by, where xy = r², is (r, ab >0)

a)

2ab√r

b)

2r√ab

c)

-2r√ab

d)

-2ab√r

20.

The absolute maximum value of $y=x^3-3x+2$ in $0\le x\le2$

a)

4

b)

6

c)

2

d)

0

21.

The absolute minimum value of $x^4-x^2-2x+5$

a)

5

b)

3

c)

7

d)

does not exist

22.

If the position of a particle is represented by x(t) = -t² + 1, what is its instantaneous velocity at t = 1?

a)

v = 0

b)

v = 1

c)

v = -1

d)

v = -2

23.

Find the second derivative of
f(x) = x²+ e^x - cosx

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

24.

Does this graph have a maximum or minimum value?

a)

maximum

b)

minimum

c)

neither

d)

both

25.

How many extrema are in the picture?

a)

2

b)

3

c)

4

d)

5

26.

What is the equation of the line tangent to y= x²+2x-1 at x=1?

a)

y=2x

b)

y=4x-2

c)

y=4x

d)

y=2x-2

27.

The velocity of an object is given as v = 2t + 3t³. what is the acceleration of the object at t = 2 s?

a)

38 m/s/s

b)

27 m/s/s

c)

16 m/s/s

d)

49 m/s/s

28.

What is the increasing interval for the function below?

a)

$\left(-\infty,\infty\right)$

b)

$\left(-\infty,2\right]$

c)

$\left(-\infty,6\right]$

d)

$\left(-6,6\right)$

29.

What is the decreasing interval(s) on the function shown?

a)

$\left(-7,\ -6\right)\ and\ \left(-2,\ 4\right)$

b)

$\left(-7,\ 0\right)\ and\ \left(-2,\ 4\right)$

c)

$\left(-7,\ -6\right)\ and\ \left(-2,\ 0\right)$

d)

$\left(-7,\ 0\right)\ and\ \left(4,\ 5\right)$

30.

The intervals in which the function f given by f (x) = x² – 4x + 6 is increasing is

a)

$\left(-\infty,\ 2\right)$

b)

$\left(2,\ \infty\right)$

c)

$\left(-\infty,\ \infty\right)$

d)

none of these

31.

The logarithmic function is increasing on

a)

$\left(0,\ 1\right)$

b)

$\left(0,\ \infty\right)$

c)

$\left(1,\ \infty\right)$

d)

$\left(-\infty,\ \infty\right)$

32.

The interval in which $y=x^2e^{–x}$ is increasing is

a)

$\left(-2,\ 0\right)$

b)

$\left(-\infty,\ \infty\right)$

c)

$\left(2,\ \infty\right)$

d)

$\left(0,\ 2\right)$

33.

The interval on which the function f (x) = 2x³ + 9x² + 12x – 1 is decreasing is:

a)

$\left[-1,\ \infty\right)$

b)

$\left[-2,\ -1\right]$

c)

$\left(-\infty,\ -2\right]$

d)

$\left[-1,\ 1\right]$

34.

Which of the following functions is decreasing on $\left(0,\ \frac{\pi}{2}\right)$

a)

$\sin2x$

b)

$\tan x$

c)

$\cos x$

d)

$\cos3x$

35.

The interval in which $y=x^2e^{–x}$ is increasing is

a)

$\left(-2,\ 0\right)$

b)

$\left(-\infty,\ \infty\right)$

c)

$\left(2,\ \infty\right)$

d)

$\left(0,\ 2\right)$

36.

y = x (x – 3)² decreases for the values of x given by :

a)

$1<x<3$

b)

$x<0$

c)

$x>0$

d)

$0<x<\frac{3}{2}$

37.

If $\frac{\text{d}x}{\text{d}y}=\infty$ , then the tangent at a point P(x,y) on the curve is perpendicular to the x- axis.

a)

False

b)

True

38.

For Maximum point : (i) $\frac{\text{d}x}{\text{d}y}=0$ ; (ii) $\frac{d^2y}{dx^2}$ < 0

a)

False

b)

True

39.

The smallest value of the polynomial

x^3-18x^2+96x

in [0, 9] is

a)

126

b)

0

c)

135

d)

160

40.

The maximum value of

\sin x.\cos x

is

a)

1/4

b)

1/2

c)

$\sqrt{2}$

d)

$2\sqrt{2}$

41.

The absolute maximum and the absolute minimum values of the function $f\left(x\right)=4x-\frac{x^2}{2}$ in [-2, 4.5] is

a)

8, -10

b)

4, - 2

c)

0, 3

d)

3, 6

42.

The points at which the function changes its nature from increasing to decreasing or from decreasing to increasing are called

(a)

43.

The slope of the normal to the curve $x=a\cos^3\theta,\ \ y=a\sin^3\theta$ at $\theta=\frac{\pi}{4}$ is

(a)

44.

The equation of the tangent line to the curve y = x² – 2x +7 which is parallel to the line 2x – y + 9 = 0 is

a)

$y\ –\ 2x\ –\ 3=0$

b)

$y\ +2x\ –\ 3=0$

c)

$y\ –\ 2x\ +\ 3=0$

d)

$y\ +2x\ +\ 3=0$

45.

The equation of normal to the curve y = tan x at (0, 0) is ________

a)

$x-y=0$

b)

$x+y=1$

c)

$x+y=0$

d)

$x-y=1$

46.

The equation of the normal to the curve $y=x^4–\ 6x^3+13x^2\ –\ 10x+5$ at $\left(0,\ 5\right)$ is

a)

$x\ \ +10y+50=0$

b)

$x\ –\ 10y+50=0$

c)

$x\ –\ 10y-50=0$

d)

$x\ +10y+50=0$

47.

The equations of all lines having slope 0 which are tangent to the curve $y=\frac{1}{x^2-2x+3}$ is

a)

$y=1$

b)

$2y=1$

c)

$2y+1=0$

d)

$y+1=0$

48.

Determine the point on the curve f(x)=(2x-1)² where the tangent is parallel to the line y=4x-2

a)

(0 , 1)

b)

(1 , 9)

c)

(1/2 , 0)

d)

(1 , 1)

49.

What is the equation of the normal line to the curve $y=3x^2+2x$ at $x=1$ ?

a)

$y=8x-3$

b)

$y=8x-13$

c)

$y=-\frac{1}{8}x+\frac{41}{8}$

d)

$y=\frac{1}{8}x+\frac{39}{8}$

50.

The point on the curve y²=xat which the tangent is parallel to the line joining (1,4) and (3,2) is

a)

(5,6)

b)

(1/2,-1/2)

c)

(1/2,3/2)

d)

(-1/2,1/2)

51.

What is the equation of the line tangent to the curve y = x² at x = 1?

a)

y = 2x + 1

b)

y = 2x - 1

c)

y = -2x + 1

d)

y = x - 1

52.

Find the equation of the normal line at x = 0 of:
$f\left(x\right)\ =\ x^3+e^x$

a)

$y=-x+1$

b)

$y=\ x+1$

c)

$y=-x\ -1$

d)

$y=x\ -\ 1$

53.

Find the derivative of the given equation
f(x) = 7

a)

7

b)

0

c)

7x

d)

14

54.

Find the derivative of the given equation
g(x) = 6 + ⅔x

a)

6

b)

12

c)

6 + ⅔

d)

⅔

55.

Find the derivative of the given equation
f(x) = 1/x²(hint: use your pink sheet)

a)

1/2x

b)

-2x

c)

2x

d)

-2x^-3

56.

Find the derivative of the given equation
f(x) = x⁴ + 4x³- 2x²

a)

x³ + x²- x

b)

4x³ + 12x²+ 4x

c)

4x + 12x - 4x

d)

4x³ + 12x²- 4x

57.

Which if the following is not a correct notation for 'derivative?'

a)

f'(x)

b)

y'

c)

x'

d)

dy/dx

58.

If a function has a derivative that is negative, what does that tell you?

a)

The function is increasing

b)

The function is decreasing

59.

a)

max is 4 at x = 0

b)

max is 4 at x = 3

c)

max is 3 at x = 5

d)

No maximum

60.

If f"(x) >0, what is true about f(x)?

a)

it is concave up there

b)

it has an inflection point

c)

It's zero

d)

it is concave down there

61.

What will be true at an inflection point?

a)

f(x)=0

b)

f'(x)=0

c)

f ''(x)=0

d)

The function is undefined

62.

Over what interval(s) is f(x) decreasing?

a)

(-3, 1)

b)

(-∞, -5) ∪ (0, 2)

c)

(-∞, -3) ∪ (1, ∞)

d)

(-5, 0) ∪ (2, ∞)

63.

Over what interval(s) is f(x) increasing?

a)

before -3 and after 1

b)

between -3 and 1

c)

(-5, 0) ∪ (2, ∞)

d)

(-5, ∞)

64.

If $x=0.5$ is a critical point of $f\left(x\right)$ and $f''\left(x\right)\ =\ \cos^2x\cdot\ln x$ then $x=0.5$ is a

a)

local maximum

b)

local minimum

c)

neither

65.

Which of the following is NOT a critical point of $f\left(x\right)\ =\ e^x\cdot x^2$ ?

a)

$x=0$

b)

$x=1$

c)

$x=-2$

66.

Where is the point of inflection for the function $f\left(x\right)\ =\ x^{3\ }+6x^2$ ?

a)

$x=0$

b)

$x=-4$

c)

$x=-2$

d)

$x=4$

67.

On what interval(s) is the function
$f\left(x\right)\ =\ x^{3\ }+6x^2$
concave up?

a)

$\left(-\infty,\infty\right)$

b)

$\left(-\infty,-2\right)$

c)

$\left(-2,\infty\right)$

d)

$\left(0,\infty\right)$

68.

The following sign chart shows whether $f'\left(x\right)$ is positive, negative, or zero.
There is/are ...

a)

a local maximum at $x=-2$

b)

a local maximum at $x=4$

c)

local maxima at $x=-2$ and $x=4$

d)

no local extrema

69.

If $f\left(x\right)\ =\ \sqrt{1-x^2}$ ,

then $f'\left(1\right)$ is...

a)

$f\left(x\right)$ is not defined at $x=1$

b)

$1$

c)

$\frac{2}{\sqrt{3}}$

d)

$f'\left(1\right)\$ does not exist

70.

The tangent line to f(x)=x³+kx² at x = 1 is parallel to the line containing points (2, 9) and (3, 10). What is the value of k?

a)

-2

b)

-1

c)

1

d)

2

71.

What is the equation of the line tangent to y= x²+2x-1 at x=1?

a)

y=2x

b)

y=4x-2

c)

y=4x

d)

y=2x-2

72.

The interval in which

y=x^2e^{-x}

is increasing is

a)

(A) $\left(-\infty,\ \infty\right)$

b)

(B) $\left(-2,\ 0\right)$

c)

(C) $\left(2,\ \infty\right)$

d)

(D) $\left(0,\ 2\right)$

73.

The line

y=x+1

is a tangent to the curve

y^2=4x\

at the point

a)

(A) $\left(1,\ 2\right)$

b)

(B) $\left(2,\ 1\right)$

c)

(C) $\left(1,\ -2\right)$

d)

(D) $\left(-1,\ 2\right)$

74.

The slope of the tangent to the curve
$x=t^2+3t-8,\ y=2t^{2\ }-2t-5$ at the point (2, -1) is

a)

(A) 22/7

b)

(B) 6/7

c)

(C) 7/6

d)

(D) -6/7

75.

The line $y=mx+1$ is a tangent to the curve $y^2=4x$ if the value of m is

a)

(A) 1

b)

(B) 2

c)

(C) 3

d)

(D) 1/2

76.

The minimum value of $\frac{x}{\log_ex}$ is

a)

(A) e

b)

(B) 1/e

c)

(C) 1

d)

(D) None of these

77.

Slope of a line parallel to x-axis is (a) .

78.

Point which is neither a point of local maxima nor a point of local minima is called (a)

79.

If the angle of intersection of two curves is a right angle, then two curves are called (a)

80.

Product of slope of a tangent and slope of a normal at a certain point on the given curve is equal to (a)

81.

A function f(x) is said to be (a) on (a, b), if $x_1<x_{2\ }\Longrightarrow\ f\left(x_1\right)>f\left(x_2\right)\ \forall\ x_1,\ x_{2\ }\in\left(a,\ b\right)$

82.

A particle moves in a straight line with a velocity of v(t) = 2t². How far does the particle move between times t = 1 and t = 3?

a)

18

b)

52/3

c)

26/3

d)

16

83.

For times t > 0, a particle moves on the x axis with its acceleration defined by a(t) = 6t – 2. If the velocity of the particle is -7 at t = 1, then at what time is the particle at rest?

a)

t = 4

b)

t = 1

c)

t = 2

d)

t = 4/3

84.

If $x=0.5$ is a critical point of $f\left(x\right)$ and $f''\left(x\right)\ =\ \cos^2x\cdot\ln x$ then $x=0.5$ is a

a)

local maximum

b)

local minimum

c)

neither

85.

Which of the following is NOT a critical point of $f\left(x\right)\ =\ e^x\cdot x^2$ ?

a)

$x=0$

b)

$x=1$

c)

$x=-2$

86.

How many critical points does the function $f\left(x\right)\ =\ e^x\cdot x^3$ have?

a)

$0$

b)

$1$

c)

$2$

d)

$\infty$

87.

Where does the global maximum of $f\left(x\right)\ =\ e^{-x^2}$ occur on $\left[-1,2\right]$ ?

a)

$x=-1$

b)

$x=0$

c)

$x=1$

d)

$x=2$

88.

The function $f\left(x\right)\ =\ \left|x-3\right|-4$ is...

a)

not continuous on $\left[-1,4\right]$

b)

not differentiable on $\left[-1,4\right]$

c)

continuous and differentiable on $\left[-1,4\right]$

d)

... what do those bars mean again?

89.

The function $f\left(x\right)\ =\ x^{\frac{2}{3}}$ is...

a)

not continuous on $\left[5,6\right]$

b)

not differentiable on $\left[5,6\right]$

c)

continuous and differentiable on $\left[5,6\right]$

d)

one of the greatest functions of all time

90.

Where is the point of inflection for the function $f\left(x\right)\ =\ x^{3\ }+6x^2$ ?

a)

$x=0$

b)

$x=-4$

c)

$x=-2$

d)

$x=4$

91.

On what interval(s) is the function
$f\left(x\right)\ =\ x^{3\ }+6x^2$
concave up?

a)

$\left(-\infty,\infty\right)$

b)

$\left(-\infty,-2\right)$

c)

$\left(-2,\infty\right)$

d)

$\left(0,\infty\right)$

92.

If $f\left(x\right)\ =\ \sqrt{1-x^2}$ ,

then $f'\left(1\right)$ is...

a)

$f\left(x\right)$ is not defined at $x=1$

b)

$1$

c)

$\frac{2}{\sqrt{3}}$

d)

$f'\left(1\right)\$ does not exist

93.

Find the inflection point(s) of $f\left(x\right)\ =\ e^x+2x^2-3x$ .

a)

$x=\frac{3}{2}$

b)

$x=0,\ x=\frac{3}{2}$

c)

$x=0$

d)

$f\left(x\right)$ has no inflection points.

94.

Which of the following functions does not satisfy the conditions put forth in Rolle's Theorem?

a)

$f\left(x\right)\ =\ \left(x-2\right)^{\frac{2}{3}}\$ on $\left[3,5\right]$

b)

$f\left(x\right)\ =\ \sqrt{1-x^2}$ on $\left[-1,1\right]$

c)

$f\left(x\right)=\left(2x-3\right)^4$ on $\left[1,2\right]$

d)

$f\left(x\right)=e^{-x^2}$ on $\left[-4,4\right]$

95.

Identify the conclusion of Rolle's Theorem.

a)

$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

b)

$f'\left(c\right)=0$

96.

Identify the conclusion of the Mean Value Theorem.

a)

$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

b)

$f'\left(c\right)=0$

97.

Where is this decreasing?

a)

(-∞, -2.55) U (0, 2.55)

b)

(+∞, -6.25) U (36, -6.25)

c)

(-∞, ∞)

d)

(-∞, 0)

98.

What are all values of x for which the function f defined by f(x)=(x² - 3)e^-x is increasing? (No calculator.)

a)

x < -1 and x > 3

b)

-3 < x < 1

c)

-1 < x < 3

d)

All values of x

99.

Find the interval of decrease for the function $y=-x^2-2x$ .

a)

$x>2$

b)

$x<2$

c)

$x>0$

d)

$x<0$

100.

Find the interval of increase and decrease for the function $y=4x^3-15x^2-72x-8$ .

a)

$-4<x<\frac{3}{2}$

b)

$-\frac{3}{2}<x<4$

c)

$x<-\frac{3}{2}\ \cup x>4$

d)

$x<-4\cup x>\frac{3}{2}$

101.

Given $f\left(x\right)=3x^5-2x^4+5x^3$ . Find $\frac{d^2y}{dx^2},\ and\ \frac{d^3y}{dx^3}$ .

a)

it is concave up there

b)

it has an inflection point

c)

It's zero

d)

it is concave down there

102.

1. The total revenue in ₹ received from the sale of x units of an article is given by R(x) = 3x² + 36x + 5. The marginal revenue when x = 15 is (in ₹ )

(a) 126

(b) 116

(c) 96

(d) 90

a)

A

b)

B

c)

C

d)

D

103.

4. The equation of the normal to the curve y = sin x at (0, 0) is

(a) x = 0

(b) y = 0

(c) x + y = 0

(d) x – y = 0

a)

A

b)

B

c)

C

d)

D

104.

What is a horizontal tangent?

a)

Has a positive slope.

b)

Has a negative slope.

c)

Has a slope of zero.

d)

Has an undefined slope.

105.

What is the slope of the line tangent to the graph of y=ln(2x) at the point where x=4?

a)

1/8

b)

1/4

c)

1/2

d)

3/4

e)

4

106.

Find the slope of the tangent line to f(x) = -3x²-6x at x = 1.

a)

m = 0

b)

f'(x) = -6x - 6

c)

f'(x) = 6x

d)

m = -12

107.

What is the equation of the line tangent to y= x²+2x-1 at x=1?

a)

y=2x

b)

y=4x-2

c)

y=4x

d)

y=2x-2

108.

What are the intervals of the graph increasing for f(x) = 2x⁴- 4x² + 1

a)

(-1,0)

b)

(0,1)

c)

(-∞,-1) and (1,∞)

d)

(-1,0) and (1,∞)

109.

Over what interval(s) is f(x) decreasing?

a)

(-3, 1)

b)

(-∞, -5) ∪ (0, 2)

c)

(-∞, -3) ∪ (1, ∞)

d)

(-5, 0) ∪ (2, ∞)

110.

Find the interval(s) where f(x)=6x²-x³ is increasing.

a)

(-4,0)

b)

(0,4)

c)

(-∞, 0)∪(4, ∞)

d)

(-∞, -4 )∪(0, ∞)

APPLICATION OF DERIVATIVES

Total questions: 110

Worksheet time: 4hrs 40mins

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