Free Practice Quiz Question List

4 Applications of Derivatives Online Quiz Questions

Use this free practice quiz with 20 questions to review 4 Applications of Derivatives, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: In a related-rates problem, it is generally correct to substitute the given numerical values before differentiating the equation relating the changing quantities.

A
True
B
False

Choose one

1 point

Suppose $f'(c)=0$ and $f''(c)>0$ . What conclusion does the second derivative test support about $f$ at $c$ ?

A
A local maximum
B
A local minimum
C
An inflection point
D
No conclusion is possible

Choose one

1 point

Use the linearization of $f(x)=\sqrt{x}$ at $a=4$ to approximate $\sqrt{4.1}$ .

A
2.0025
B
2.0125
C
2.025
D
2.05

Written response

1 point

For $f(x)=x^2$ on $[2,4]$ , enter the value of $c\in(2,4)$ guaranteed by the Mean Value Theorem such that $f'(c)$ equals the average rate of change on the interval.

Fill in the blank

1 point

Complete the two hypotheses of the Mean Value Theorem: If $f$ is on $[a,b]$ and on $(a,b)$ , then there is at least one $c\in(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$ .

True or false

1 point

True or false: Every critical point of a differentiable function is automatically a local maximum or a local minimum.

A
True
B
False

Written response

1 point

A spherical balloon is filled at $\frac{dV}{dt}=2\text{ cm}^3/\text{s}$ . When its radius is $3\text{ cm}$ , how fast is the radius increasing? Enter the rate in cm/s rounded to four decimal places. Answers within an absolute tolerance of 0.0001 cm/s are accepted.

Choose all

1 point

Which steps are part of a valid procedure for finding the absolute maximum and minimum of a function that is continuous on $[a,b]$ ? Select all correct choices.

A
Find critical points in the open interval.
B
Evaluate the function at both endpoints.
C
Assume the largest critical-point value is the absolute maximum without checking endpoints.
D
Compare all critical-point and endpoint values.

Fill in the blank

1 point

Complete the concavity rules: If $f''(x)$ is on an interval, the graph is concave up; if $f''(x)$ is on an interval, the graph is concave down.

Choose all

1 point

For motion along a line, which statements are correct? Select all correct choices.

A
If $v(t)>0$ , the object moves forward.
B
If $v(t)<0$ , the object moves backward.
C
If $v(t)>0$ and $a(t)>0$ , the object speeds up.
D
If $v(t)<0$ and $a(t)<0$ , the object speeds up.

Choose one

1 point

Apply one iteration of Newton's method to $f(x)=x^2-2$ starting from $x_0=1$ . What is $x_1$ ?

A
0.5
B
1
C
1.5
D
2

Choose one

1 point

In marginal analysis, which equation gives the usual interior critical-point condition for maximizing profit, where $P(x)=R(x)-C(x)$ ?

A
Average revenue equals average cost
B
Marginal revenue equals marginal cost
C
Revenue equals zero
D
Marginal cost equals zero

Open ended

1 point

For $f(x)=x^4-4x^2$ , determine all critical points and classify each local maximum or minimum. Then determine all inflection points, verifying the required concavity changes. Give exact coordinates and justify your conclusions with derivative sign information.

Choose one

1 point

In a related-rates problem, which procedure is mathematically appropriate?

A
Substitute every known numerical value before writing an equation.
B
Differentiate the relationship with respect to time, then substitute the known values.
C
Solve for the variables first and differentiate only the final numerical equation.
D
Differentiate only the quantity whose rate is being requested.

Written response

1 point

In an optimization problem, what is the standard term for the quantity being maximized or minimized?

Choose one

1 point

A company can sell $x$ units at a price of $p(x)=100-2x$ dollars per unit. At what sales level is the revenue maximized?

A
10 units
B
20 units
C
25 units
D
50 units

Choose one

1 point

For $f(x)=\frac{x+1}{x-2}$ , which statement about the graph is correct?

A
The vertical asymptote is $x=-1$ .
B
The vertical asymptote is $x=2$ .
C
The horizontal asymptote is $y=2$ .
D
There are no asymptotes because the function is rational.

Choose one

1 point

A manufacturer's cost is $C(x)=x^2-20x+300$ dollars for producing $x$ units. For what production level is the cost minimized?

A
10 units
B
20 units
C
100 units
D
300 units

True or false

1 point

True or false: If $f'(x)>0$ at every point of an interval, then $f$ is increasing on that interval.

A
True
B
False

Written response

1 point

A rectangle has a perimeter of $40$ metres. What is its maximum possible area? Enter the value in square metres.