Calculus Unit 4 Review

What is the first derivative test used for?

• Determining the concavity of a function

• Finding the area under a curve

• Determining local extrema of a function

• Solving for the x-intercepts of a function

State the second derivative test for finding the nature of critical points.

• If the second derivative of a function at a critical point is positive, then the function has a local minimum at that point. If the second derivative is negative, then the function has a local maximum at that point. If the second derivative is zero, the test is inconclusive.

• If the second derivative is negative, then the function has a local minimum at that point.

• If the second derivative is positive, then the function has a local maximum at that point.

• If the second derivative is positive, then the function has a local minimum at that point.

Which is the best description of Newton's method?

• Newton's method can only be used to find roots of quadratic equations.

• Newton's method is an iterative method for finding the roots of a real-valued function.

• Newton's method is not applicable in calculus.

• Newton's method is a non-iterative method for finding the roots of a real-valued function.

What is the main difference between the first and second derivative test?

• Finding the maximum vs. finding the minimum

• Determining slope vs. determining curvature

• Identifying points of inflection vs. identifying local extrema

• Determining increasing and decreasing behavior vs. determining concavity and points of inflection

How is the Mean Value Theorem related to the concept of average rate of change?

• It has no relationship to the concept of average rate of change

• It only applies to linear functions, not average rate of change

• It is only used for finding the maximum value of a function, not average rate of change

• It provides a precise mathematical relationship between the average rate of change of a function over an interval and the instantaneous rate of change at a specific point within that interval.

Discuss a real-life scenario where related rates are applicable and explain the problem-solving approach.

• Baking a cake

• Water tank filling or emptying

• Planting a garden

• Driving a car

Explain the concept of concavity and how it is related to the second derivative test.

• The second derivative test is used to determine the concavity of a function at a given point

• Concavity is related to the first derivative test and has no connection to the second derivative

• Concavity is only related to the first derivative and has no impact on the second derivative

• Concavity is related to the second derivative test as it helps to determine whether a function is concave up or concave down at a critical point. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down.

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

• 3 ft/sec
• -7.2 ft/sec
• 7.2 ft/sec
• 12

Brandon is starting to clean up after a birthday party. He begins deflating each spherical balloon by puncturing a hole in each. The air leaves the balloon at a constant rate of 2 cm³/sec.  How fast is the diameter changing when the diameter is 8 cm?

• -1/(16𝜋) cm/sec

• -1/16 cm/sec
• -1/(4𝜋) cm/sec

• 1/(4𝜋) cm/sec

Louisa and Karis were each dropped off at the same bus stop. Louisa’s bus drops her off at 3:30 whereas Karis is dropped off ten minutes later. Louisa runs home at a constant rate of 6 mph and Karis runs home at 3 mph. Louisa lives north of the bus stop and Karis lives to the east.  How fast is the distance between them changing at 4:00?

• 6.512 mph
• 7.115 mph
• 6.708 mph
• 6.641 mph