Free Printable Quiz 2 (weeks 5-7)

1.

What does it mean that a function has an absolute maximum at point c?

a)

f(c) < f(x) for all x

b)

f(c) ≥ f(x) for all x

c)

f′(c)=0

d)

f′′(c)<0

2.

Which condition is required to apply the Extreme Value Theorem?

a)

The function is differentiable

b)

The interval is open

c)

The function is continuous on a closed interval

d)

The derivative is never zero

3.

A critical point of a function is a point where

a)

f(x)=0

b)

the function has a maximum

c)

f′(x)=0 or f′(x) does not exist

d)

the graph crosses the x-axis

4.

According to Rolle’s Theorem, if f(a)=f(b), then

a)

f′(x)=0 for all x

b)

there exists c∈(a,b) such that f′(c)=0

c)

the function is constant

d)

f′′(c)=0

5.

The Mean Value Theorem states that

a)

the average value of a function is zero

b)

the function has an extremum

c)

the derivative is always positive

d)

there exists a point where the tangent line is parallel to the secant line

6.

If f′(x)>0 on an interval (a,b), then the function ... .

a)

is decreasing

b)

has a maximum

c)

is increasing

d)

is constant

7.

All antiderivatives of the same function differ by

a)

a sign

b)

a variable

c)

a coefficient

d)

a constant

8.

$\int3x^2dx=$

a)

$x^3$

b)

$x^3+C$

c)

$3x+C$

d)

$6x+C$

9.

Which method is best suited for evaluating the integral $\int2x\cos\left(x^2\right)dx?$

a)

Integration by parts

b)

Substitution

c)

Partial fractions

d)

Trigonometric substitution

10.

For trigonometric substitution involving $\sqrt[]{a^2-x^2}$ , the usual substitution is

a)

x=asinθ

b)

x=atanθ

c)

x=asec⁡θ

d)

x=acosθ

11.

A global maximum of a function

a)

must occur at a critical point

b)

can occur at an endpoint of a closed interval

c)

only occurs where f′(x)=0

d)

cannot be a local maximum

12.

Which of the following guarantees the existence of both an absolute maximum and minimum?

a)

Continuity on an open interval

b)

Differentiability on a closed interval

c)

Continuity on a closed, bounded interval

d)

Existence of critical points

13.

Fermat’s Theorem applies only if the function

a)

is differentiable at the point

b)

is continuous

c)

is defined on an open interval

d)

has two equal values

14.

The indefinite integral of a function represents

a)

a single function

b)

the area under the curve

c)

all antiderivatives of the function

d)

the derivative of the function

15.

$\int\cos xdx=$

a)

−sinx+C

b)

sinx+C

c)

tan⁡x+C

d)

secx+C

16.

$\int a^xdx=$

a)

ln⁡x+C

b)

$\frac{a^x}{\ln a}+C$

c)

$e^x+C$

d)

$xa^x+C$

17.

$\int\sec x\tan x\ dx\ =$

a)

$\sec x+C$

b)

$\tan x+C$

c)

$\cos x+C$

d)

$\sec x\tan x+C$

18.

$\int\frac{dx}{a^2+x^2}dx=$

a)

$\sinh^{-1}\left(\frac{x}{a}\right)+C$

b)

$\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C$

c)

$\sin^{-1}\left(\frac{x}{a}\right)+C$

d)

$\frac{1}{a}\tanh^{-1}\left(\frac{x}{a}\right)+C$

19.

Integration by parts is based on which rule?

a)

Chain rule

b)

Quotient rule

c)

Product rule

d)

Power rule

20.

Substitution is most useful when

a)

the integrand contains a composite function

b)

the integrand is a rational function

c)

trigonometric identities are needed

d)

limits are infinite

21.

Partial fractions are used to integrate

a)

trigonometric functions

b)

exponential functions

c)

logarithmic functions

d)

rational functions

22.

For integrals involving $\sqrt[]{a^2+x^2}$ , the standard substitution is

a)

x = a sinθ

b)

x = a tan⁡θ

c)

x = a secθ

d)

x = a cosθ

23.

Before using partial fractions, the degree of the numerator must be

a)

greater than the denominator

b)

equal to the denominator

c)

less than the denominator

d)

irrelevant

24.

Trigonometric substitution works because

a)

trigonometric functions are periodic

b)

radicals can be simplified using identities

c)

derivatives disappear

d)

limits become zero

25.

$\int\ln xdx=$

a)

$x\ln x-x+C$

b)

$x\ln x+x+C$

c)

$x\ln x+C$

d)

$\ln x-x+C$

26.

What is the result of the integral $\int e^xdx=$

a)

$e^x-x+C$

b)

$e^x+x+C$

c)

$xe^x+C$ x

d)

$e^x+C$

27.

For the integral $\int\frac{1}{x}dx=$ , the result is

a)

e^x+C

b)

1/x+C

c)

ln|x|+C

d)

ln(x)+C

28.

What is the derivative of $\sin x$ ?

a)

- cos x

b)

- sin x

c)

cos x

d)

sin x

29.

Which of the following is a condition for a function to be continuous at a point?

a)

The limit must exist at that point

b)

The limit must equal the function value at that point

c)

All of the above

d)

It must be defined at that point

30.

What is the derivative of $\ln x$ ?

a)

e^x

b)

x

c)

ln x

d)

1/x

Quiz 2 (weeks 5-7)

Total questions: 30

Worksheet time: 15mins

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