Free Printable Chapter 16 - Applications of Differentiation

1.

If (a,b) is a local minimum, then what will be true about f'(a)?

a)

It's positive

b)

It's negative

c)

It's zero

d)

Cannot be determined

2.

Given a function, f(x), if f'(x)<0 over a certain interval, then f(x) is ____________ over that interval.

a)

decreasing

b)

increasing

c)

concave up

d)

concave down

3.

The slope of a function is described by its ____________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

4.

What is the stationary point for the curve y=x²- 4?

a)

A minimum at ( 0, -4)

b)

A maximum at (0, -4)

c)

A minimum at (0,4)

d)

A maximum at (0,4)

5.

What is the stationary points for the curve f(x) = x³ +3x² - 24x and determine their nature.

a)

Maximum point : (2, -28) and minimum point : (-4, 80)

b)

Minimum point : (2, 28) and maximum point : (-4, 80)

c)

Minimum point : (2, -28) and maximum point : (4, 80)

d)

Minimum point : (2, -28) and maximum point : (-4, 80)

6.

Which of the following is the best definition of REVENUE?

a)

The total amount of income a business makes from selling products or services

b)

The amount of money a business has LEFT over after paying for their costs.

c)

The total amount of money a business spends.

d)

The amount of money a business loses every month.

7.

Which of the following is the best definition of PROFIT?

a)

The total amount of income a business makes from selling products or services

b)

The amount of money a business has left over after paying for their costs.

c)

The total amount of money a business spends.

8.

Which of the following is the equation for figuring out PROFIT?

a)

Profit = Revenue - Cost

b)

Profit = Cost - Revenue

c)

Revenue = Profit - Cost

d)

Costs = Profit - Revenue

9.

Concave downward is obtained when

a)

f '' is negative

b)

f ' is positive

c)

f ' is negative

d)

f '' is positive

10.

A spherical ball is being inflated at a rate of 20 $\frac{cm^3}{s}$ . Find the rate of increase of its radius when it's surface area is $100\pi\ cm^2$

a)

$\frac{1}{6\pi}$ cm/s

b)

$\frac{1}{4\pi}$ cm/s

c)

$\frac{1}{5\pi}$ cm/s

d)

$\frac{2}{5\pi}$ cm/s

11.

The radius of a sphere is decreasing at a rate of 2 cm/sec. At the instant when the radius of the sphere is 3 cm, what is the rate of change, in cm²/sec, of the surface area of the sphere? (The surface area of a sphere is A = 4πr².)

a)

-108π

b)

-72π

c)

-48π

d)

-24π

12.

What is the turning point for the curve y=-3x²-12x+5 ?

a)

A minimum at (-2, 17)

b)

A maximum at (-2, 17)

c)

A minimum at (2, -31)

d)

A maximum at (2, -31)

13.

What is the stationary point for the curve y=x²- 4?

a)

A minimum at ( 0, -4)

b)

A maximum at (0, -4)

c)

A minimum at (0,4)

d)

A maximum at (0,4)

14.

What is the stationary points for the curve f(x) = x³ +3x² - 24x and determine their nature.

a)

Maximum point : (2, -28) and minimum point : (-4, 80)

b)

Minimum point : (2, 28) and maximum point : (-4, 80)

c)

Minimum point : (2, -28) and maximum point : (4, 80)

d)

Minimum point : (2, -28) and maximum point : (-4, 80)

15.

$If\ v=t^3+2t^2+3t+1,\ calculate\ the\ value\ of\ \frac{\text{d}^2v}{\text{d}t^2}\ \ \ when\ t=\frac{2}{3}$

a)

$\frac{\text{d}^2v}{\text{d}t^2}=100$

b)

$\frac{\text{d}^2v}{\text{d}t^2}=-20$

c)

$\frac{\text{d}^2v}{\text{d}t^2}=-10$

d)

$\frac{\text{d}^2v}{\text{d}t^2}=10$

16.

if

y=ax^{\frac{5}{4}},\

where a is a constant , find the approximate percentage change in x when there is a 5% change in y.

a)

4%

b)

5%

c)

-4%

d)

6%

17.

Find the equation of the normal to the curve $y=x^3-2x^2+3$ at the point x=2

a)

4y+x=14

b)

4y+3x=14

c)

y+x=14

d)

4y+x=41

18.

The volume of the sphere is given by $v=\frac{4}{3}\pi r^3$ ,where r is the radius. If there is 1.5% change in it's radius find the corresponding percentage change in it's volume.

a)

4%

b)

5%

c)

4.5%

d)

5.5%

19.

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

20.

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)

$x=4$

21.

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

22.

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

a)

$0$

b)

$1$

c)

$2$

d)

$\infty$

Chapter 16 - Applications of Differentiation

Total questions: 22

Worksheet time: 6hrs 30mins

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