AP Calculus Unit 4 Review

Front

What is the formula for the volume of a sphere?

Back

V = \frac{4}{3} \pi r^3

Tags

CCSS.8.G.C.9

CCSS.HSG.GMD.A.3

Front

How do you find the rate of change of volume with respect to time for a spherical balloon?

Back

Use the formula \frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt} and apply the chain rule.

Front

What is the derivative of the volume of a sphere with respect to its radius?

Back

\frac{dV}{dr} = 4\pi r^2

Front

If the radius of a spherical balloon is decreasing, how does it affect the volume?

Back

The volume decreases as the radius decreases.

Front

What is the relationship between position, velocity, and acceleration in calculus?

Back

Acceleration is the derivative of velocity, and velocity is the derivative of position.

Front

How do you find the acceleration of an object given its position function?

Back

Take the second derivative of the position function with respect to time.

Front

What is linear approximation in calculus?

Back

Linear approximation uses the tangent line at a point to estimate the value of a function near that point.

Front

How do you perform a linear approximation for a function f(x) at a point a?

Back

f(a + h) \approx f(a) + f'(a) \cdot h, where h is a small change in x.

Front

What is the limit of a function, and why is it important in calculus?

Back

The limit describes the behavior of a function as it approaches a certain point, which is essential for understanding continuity and derivatives.

Front

How do you evaluate limits involving trigonometric functions?

Back

Use algebraic manipulation, L'Hôpital's rule, or known limit properties.