Free Printable Differentiation and AOD

1.

. y = x³ – 2x² + 5x + 8. The gradient of the curve at x = 2 is

a)

9

b)

8

c)

15

d)

25

2.

The two natural numbers whose sum is 30 and product is a minimum are,

a)

10, 20

b)

12, 18

c)

15, 15

d)

14, 16

3.

The length of seg AB is 12 cm. The point P on seg AB such that AP² + BP² is minimum is,

a)

AP = 5, BP = 7

b)

AP = 6, BP = 6

c)

AP = 8, BP = 4

d)

AP = 9, BP = 3

4.

Consider the function

P (x) = 2 + 3x² + 5x⁴ + x⁶. Then P (x) has

a)

Neither a maximum nor a minimum

b)

only one maximum

c)

Only one minimum

d)

Only one maximum and one minimum

5.

A man is walking at the rate of 6.5km/hr.toward the foot of a tower 120m. high at what rate is he approaching the top of the tower when he is 50m. away from the tower ?

a)

2.5 km/hr.

b)

0.25km/hr.

c)

2.5m/sec.

d)

none

6.

The volume of a cube is increasing at a rate of 7cm³/sec. How fast is the surface area increasing when the length of an edge is 12cm.?

a)

7/3 cm²/sec.

b)

3/7 cm²/sec.

c)

7/3 cm/sec.

d)

7/3 cm³/sec.

7.

Approximate value of 6x³-7x²+x-3 when x =2.01 is

a)

20

b)

19.50

c)

19.45

d)

19.54

8.

The function f (x) = - x² – 2x + 15 is increasing in the interval

a)

(- ∞, - 1)

b)

(- 1, ∞)

c)

(- 1, 1)

d)

none

9.

Local maximum value of the function logx/x is

a)

1

b)

e

c)

1/e

d)

none

10.

Approximate value of $\log_{10}\left(103\right)$

a)

2.00121

b)

2.001302

c)

2.00421

d)

2.00142

11.

The approximate value of tan(29⁰30’)

a)

0.6005

b)

0.6007

c)

0.4005

d)

none

12.

Equation of normal to the curve

2x² – y²=14 which is parallel to x +3y = 4 is

a)

x +3y=7

b)

x+3y-9=0

c)

2x +3y=4

d)

none

13.

The tangent to the parabola x² = 2y at the point (1, 1/2) makes with the x –axis an angle of

a)

0⁰

b)

45⁰

c)

60⁰

d)

90⁰

14.

A stone thrown vertically upwards satisfies the equation x = 80t – 16t². The time required to reach the maximum height is

a)

2

b)

3

c)

4

d)

2.4

15.

A manufacturer can sell x items at a price of Rs. (330 – x) each. The cost of production of x items is Rs. (x² + 10x + 12). For maximum profit the number of items to be sold is

a)

20

b)

40

c)

60

d)

80

16.

Let f be the function with derivative given by f'(x) = x² - 2/x. On which of the following intervals is f decreasing?

a)

(-∞, 0]

b)

(-1, 0]

c)

(0, ∛2]

d)

(∛2, ∞)

17.

A solid sphere has radius r cm, surface area A $cm^2$ and volume V $cm^3$ . The radius is increasing at a rate of $\frac{1}{5\pi}$ $cm\ s^{-1}$ . Find the rate of increase of the surface area when r=3.

a)

2.4 $cm\ s^{-1}$

b)

24 $cm\ s^{-1}$

c)

4.8 $cm\ s^{-1}$

d)

48 $cm\ s^{-1}$

18.

Find the derivative:
y=5x²e^3x

a)

y'=10xe^3x(2x+3)

b)

y'=5xe^3x(3x+2)

c)

y'=10ex^3x(3x+2)

d)

y'=5xe^3x(2x+3)

19.

Find $\frac{dy}{dx}$ for $y=\tan^4\left(2x+1\right)$ .

a)

$4\left(\tan\left(2x+1\right)\right)^4\cdot\sec^2\left(2x+1\right)\left(2\right)$

b)

$\left(\tan\left(2x+1\right)\right)^3\cdot\sec^2\left(2x+1\right)\left(2\right)$

c)

$4\left(\tan\left(2x+1\right)\right)^3\cdot\sec^2\left(2x+1\right)$

d)

$4\left(\tan\left(2x+1\right)\right)^3\cdot\sec^2\left(2x+1\right)\left(2\right)$

20.

If 2^x+2^y=2^x+y, then at x=y=1 dy/dx

a)

1

b)

2

c)

-1

d)

0

21.

If log(x+y) = 2xy, then y’ (0)=

a)

1

b)

-1

c)

2

d)

0

22.

If y=1+x.e^y, then

a)

ye⁴

b)

(2+y)e^y

c)

e^y\ /\ (2+y)\

d)

$e^y/(2-y)$

23.

$Ifx=\cos\theta\ ,\ y=\sin\theta,\ then\ \frac{dy}{dx}=?$

a)

$\cot\theta$

b)

$-\cot\theta$

c)

$\tan\theta$

d)

$-\tan\theta$

24.

$x=a(\cos t+\log\tan\frac{t}{2}),\ y=a\sin t,then\ \frac{dy}{dx}=$

a)

tan t

b)

-tan t

c)

cot t

d)

-cot t

25.

If y = f (x³), z = g (x⁵), f’ (x) = tan x and

g’ (x) = sec x, then the value of dy/dz is

a)

$\frac{3}{(5x^2)}\ \ \ \ \frac{⋅\tan⁡〖x^3〗}{\sec⁡〖x^5〗}$

b)

$\frac{(5x^2)}{3}\ \frac{⋅\sec⁡〖x^5〗}{\tan⁡〖x^3〗}$

c)

$\frac{(3x^2)⋅\tan⁡〖x^3〗}{5\sec⁡〖x^5〗}$

d)

none of these

26.

y = a^x + x^a + x^x , find dy/dx

a)

a^x log a + ax^a-1 – x.x^x-1

b)

xa^x-1 + ax^a-1 + x^x.log x

c)

a^x log a + a.x^a-1 + x^x (1 + log x)

d)

a^x + a.x^a-1 + x^x (1 + log x)

27.

If cos (x + y) = y sin x, then find dy/dx

a)

$-\frac{(\sin⁡(x+y)+y\cos⁡x)}{\left((\sin⁡x+\sin⁡(x+y)\right)}$

b)

$\frac{(\sin⁡(x+y)+y\cos⁡x)}{\left((\sin⁡x+\sin⁡(x+y)\right)}$

c)

$\ \frac{(y\cos⁡x-\sin⁡(x+y))}{(\sin⁡x-\sin⁡(x+y))}$

d)

none of these

28.

$If\ \ x^3=(x+y)^n.y^2\ \ and\ \frac{dy}{dx\ }=\frac{y}{x}\ then\ \ \ n=$

a)

1

b)

2

c)

3

d)

5

29.

$If\ y=\sqrt{\frac{\left(x-a\right)\left(x-b\right)}{\left(x-c\right)\left(x-d\right)}}\ \frac{dy}{dx}=$

a)

$\frac{y}{2}\ \ \left(\frac{1}{x-a}+\frac{1}{x-b}-\frac{1}{x-c}-\frac{1}{x-d}\right)$

b)

$\frac{y}{1}\ \ \left(\frac{1}{x-a}+\frac{1}{x-b}-\frac{1}{x-c}-\frac{1}{x-d}\right)$

c)

$\frac{1}{2}\ \ \left(\frac{1}{x-a}+\frac{1}{x-b}-\frac{1}{x-c}-\frac{1}{x-d}\right)$

d)

none of these

30.

$Ify=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+.......\infty}}}$ dy/dx=

a)

$\frac{x}{2y-1}$

b)

$\frac{x}{2y+1}$

c)

$\frac{1}{x\left(2y-1\right)}$

d)

$\frac{1}{x\left(2y+1\right)}$

31.

$Ifx^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\ \ then\ \frac{dy}{dx}=$

a)

$\left(\frac{y}{x}\right)^{\frac{1}{3\ }}$

b)

$-\left(\frac{y}{x}\right)^{\frac{1}{3\ }}$

c)

$\left(\frac{x}{y}\right)^{\frac{1}{3\ }}$

d)

$-\left(\frac{x}{y}\right)^{\frac{1}{3\ }}$

32.

$x^2+y^{2\ }=t-\frac{1}{t\ }\ ,\ x^{4\ }+y^{4\ }=t^{2\ }+\frac{1}{t^2},\ find\ x^3y\ \frac{dy}{dx}=$

a)

0

b)

1

c)

-1

d)

-3

33.

Verify Rolle’s theorem for each of the following functions on the indicated intervals :f(x) = $x\left(x-2\right)^2$ in [0,2].Find c

a)

2

b)

1

c)

1.5

d)

0

34.

$Iff(x)=e^x\sin x\ \ \ \ in\ \ [0,π],then\ \$ c(degree) in Rolle's theorem is

a)

30

b)

45

c)

135

d)

90

35.

Given an interval[a, b] that satisfies hypothesis of Rolle's theorem for the function f (x) = x^3 − 2x^2 + 3. It is known that a = 0. Find the value of b

a)

2

b)

1

c)

0

d)

3

36.

$If\ f\left(x\right)=\sqrt{x+4}$ Verify Lagrange's mean value theorem for the function on [0,5].
Write the answer as yes or no

(a)

37.

A Rectangular sheet of paper has it area 24 sq. meters. The margin at the top and the bottom are 75 cm each and at the sides 50 cm each. What are the dimensions of the paper, if the area of the printed space is maximum ?

Hint answer as (2,3)

(a)

38.

An open box is to be cut out of piece of square card coard of side 18 cm by cutting of equal squares from the corners and turning up the sides. Find the maximum volume of the box.

(a)

39.

The normal to the curve x^2 + 2xy − 3y^2 = 0 at (1, 1)

a)

Meets the curve again in second quadrant

b)

Does not meet the curve again

c)

Meets the curve again in third quadrant

d)

Meets the curve again in fourth quadrant

40.

a)

$\frac{1}{x+1}$

b)

$-\frac{1}{\left(x+1\right)^2}$

c)

$-\frac{1}{x+1}$

d)

$\left(1+x\right)^2$

41.

$Ify=\sin^{-1}⁡\ \ \left(\frac{(a\sin⁡x+b\cos⁡x)\ }{\sqrt{(a^2+b^2)}}\ \right)\ \frac{dy}{dx}=$

a)

1

b)

0

c)

$a^2+b^2$

d)

$a^{2\ }-b^2$

42.

If y = a sin ( logx) + bcos( logx), then

……

a)

$x^2\ \frac{(d^2y)}{dx^2}-x\ \ \frac{dy}{dx}-y=0$

b)

$x^2\ \frac{(d^2y)}{dx^2}-x\ \ \frac{dy}{dx}+y=0$

c)

$x^2\ \frac{(d^2y)}{dx^2}+x\ \ \frac{dy}{dx}-y=0$

d)

$x^2\ \frac{(d^2y)}{dx^2}+x\ \ \frac{dy}{dx}+y=0$

43.

$Ify=(\tan^{-1}x)^2,then\ \ (1+x^2)^2y_2^2+2xy_1(1+x^2)=…$

a)

1

b)

0

c)

2

d)

4

44.

$\left[Ify=\cos\frac{1}{2}\left(\cos^{-1}⁡x\right)\right],then$

a)

4y

b)

$\frac{4}{y}$

c)

$\frac{y}{4}$

d)

$\frac{1}{4y}$

45.

$\frac{d}{dx}\ \frac{(x^4+x^2+1)}{(x^2+x+1)}=ax+b\ \ \ then\ \ \left(a,b\right)$

a)

2,-1

b)

2,1

c)

1,2

d)

-3,1

46.

$The\ derivative\ of\sec^{-1}\left(\frac{1}{(2x^2-1)}\right)w.r.t.\ √(1-x^2)\ at\ x=½\ \ is$

a)

2

b)

4

c)

1

d)

-2

47.

If y = tan^-1( secx – tanx) , then dy/dx =

a)

1/2

b)

1

c)

-1/2

d)

-1

48.

If f(x) = log_x (log x), then f '(x) at x = e is

a)

e

b)

1/e

c)

1

d)

none

49.

The function x⁵ – 5x⁴ – 10 has a maximum when x =

a)

3

b)

2

c)

4

d)

0

50.

A square plate is contracting at the uniform rate at 2 cm²/sec. The rate at which the perimeter is decreasing when the side of the square is 16 cm long, is

a)

1/2

b)

1/4

c)

1

d)

none of these

Differentiation and AOD

Total questions: 50

Worksheet time: 32mins

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