Free Practice Quiz Question List

7. Optimization and Approximation Online Quiz Questions

Use this free practice quiz with 20 questions to review 7. Optimization and Approximation, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: If a function is continuous on a closed, bounded interval, then it must attain both an absolute maximum and an absolute minimum on that interval.

A
True
B
False

Choose one

1 point

In an optimization model, what is the main purpose of using the constraint equation?

A
Choose a numerical answer before defining any variables.
B
Use the constraint to express the objective in terms of one independent variable when possible.
C
Ignore the physical domain because the derivative determines the answer completely.
D
Replace the objective with an unrelated quantity that is easier to differentiate.

Fill in the blank

1 point

The linearization of a differentiable function f at x=a is .

Written response

1 point

For the demand equation p(q)=100-2q and cost C(q)=20q+100, how many units should be sold to maximize profit? Enter the number of units as a whole number.

Choose all

1 point

Which situations can cause Newton’s method to fail or become unusable? Select all correct choices.

A
The derivative at an iterate is zero.
B
The initial guess is written as a decimal rather than a fraction.
C
An iterate moves into an unsuitable part of the domain.
D
The sequence of approximations oscillates or diverges.
E
The function has a tangent line at every point in its domain.

True or false

1 point

True or false: Whenever f'(c)=0, the function f necessarily has a local maximum or local minimum at x=c.

A
True
B
False

Fill in the blank

1 point

For y=f(x), the differential dy is defined by dy=.

Written response

1 point

What is the absolute minimum value of f(x)=x^3-3x on the interval [-2,2]? Enter the value as an integer.

Choose all

1 point

Which steps are required when finding the absolute extrema of a continuous function on a closed interval? Select all correct choices.

A
Evaluate the function at both endpoints.
B
Find interior points where f'(x)=0.
C
Evaluate only the point with the largest x-coordinate.
D
Include interior points where f'(x) is undefined.
E
Compare all resulting function values.

Choose one

1 point

Newton’s method is applied to f(x)=x^2-10 with initial guess x₀=3. What is the next approximation x₁, rounded to seven decimal places?

A
3.0000000
B
3.1666667
C
3.3333333
D
6.0000000

Open ended

1 point

A closed cylindrical can must have fixed volume V. Explain how to use derivatives to find the radius and height that require the least material. Include the constraint, the one-variable surface-area function, the critical-point equation, the resulting dimensions, and a justification that the result is a minimum.

Choose one

1 point

If q represents the quantity sold and p(q) is the price per unit, which expression represents total revenue?

A
R(q)=q p(q)
B
R(q)=p(q)/q
C
R(q)=q+p(q)
D
R(q)=C(q)-q p(q)

True or false

1 point

True or false: If a function is continuous on a closed, bounded interval, then it must have both an absolute maximum and an absolute minimum on that interval.

A
True
B
False

Choose one

1 point

Which statement describes a constraint equation in an optimization model?

A
It identifies the quantity to be maximized or minimized.
B
It expresses a condition that the variables must satisfy.
C
It lists the physically or economically meaningful values of the variable.
D
It always gives the restriction imposed on the variables instead of measuring the quantity being optimized.

Choose one

1 point

A projectile has height $h(t)=-16t^2+64t+5$ feet. What is its maximum height?

A
1 ft
B
5 ft
C
69 ft
D
128 ft

Choose one

1 point

A rectangle has perimeter 40 meters. Which dimensions maximize its area?

A
5 m by 15 m
B
8 m by 12 m
C
9 m by 11 m
D
10 m by 10 m

Choose one

1 point

A cylindrical can has fixed volume $V$ , radius $r$ , and height $h$ . Which equation expresses $h$ in terms of $r$ and $V$ using the volume constraint?

A
$h=\frac{V}{\pi r^2}$
B
$h=\frac{V}{\pi r}$
C
$h=\frac{\pi r^2}{V}$
D
$h=\frac{V}{2\pi r^2}$

Choose one

1 point

A product has demand equation $p(q)=100-2q$ and cost $C(q)=20q+100$ . How many units should be sold to maximize profit?

A
10 units
B
20 units
C
40 units
D
60 units

Written response

1 point

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

Written response

1 point

Newton’s method is applied to $f(x)=x^2-10$ with initial guess $x_0=3$ . Using $x_{n+1}=\frac12\left(x_n+\frac{10}{x_n}\right)$ , enter the exact value of $x_1$ as a fraction.