AP Calculus Unit 4 Assessment

Front

What is the formula for the volume of a sphere?

Back

V = (4/3)πr³, where r is the radius.

Front

How do you find the rate of change of the volume of a sphere with respect to time?

Back

Use the chain rule: dV/dt = dV/dr * dr/dt.

Front

What does v(t) > 0 indicate about a particle's motion?

Back

The particle is moving to the right.

Front

What is linear approximation in calculus?

Back

It is a method to estimate the value of a function near a given point using the tangent line.

Front

How do you determine when a body is moving at a constant speed from a velocity graph?

Back

Look for intervals where the velocity is constant (horizontal line on the graph).

Front

What is the derivative of f(x) = x³ - x?

Back

f'(x) = 3x² - 1.

Front

What is the relationship between the radius and volume of a sphere when the radius is decreasing?

Back

As the radius decreases, the volume also decreases, and the rate of change can be calculated using related rates.

Front

What is the Pythagorean theorem and how is it applied in related rates problems?

Back

a² + b² = c²; it relates the sides of a right triangle, often used to find relationships between changing dimensions.

Front

How do you find the rate at which the top of a ladder is sliding down the wall?

Back

Use related rates: differentiate the Pythagorean theorem with respect to time.

Front

What does it mean for a function to be increasing?

Back

f'(x) > 0 for all x in the interval.