Unit 4-5 Test Review Practice Problems

Front

What does it mean for a function f(x) to be decreasing over an interval?

Back

A function f(x) is decreasing over an interval if, for any two points x1 and x2 in that interval where x1 < x2, the value of f(x1) > f(x2).

Front

How do you calculate the average rate of change of a function f(x) over an interval [a, b]?

Back

The average rate of change is calculated using the formula: \( \frac{f(b) - f(a)}{b - a} \).

Front

What is concavity in relation to a function's graph?

Back

Concavity refers to the direction in which a function curves. A function is concave up if its graph opens upwards (like a cup), and concave down if it opens downwards.

Front

How do you determine the intervals where a function is concave up using its first derivative f'?

Back

A function is concave up where its first derivative f' is increasing. This can be determined by finding where the second derivative f'' is positive.

Front

What is linear approximation and how is it used?

Back

Linear approximation uses the tangent line at a point to estimate the value of a function near that point. It is given by: \( f(x) \approx f(a) + f'(a)(x - a) \).

Front

What is the formula for the volume of a sphere?

Back

The volume V of a sphere with radius r is given by the formula: \( V = \frac{4}{3} \pi r^3 \).

Front

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

Back

To find the rate of change of volume with respect to radius, differentiate the volume formula: \( \frac{dV}{dr} = 4\pi r^2 \).

Front

What is the relationship between the radius of a sphere and its volume as the radius decreases?

Back

As the radius decreases, the volume of the sphere also decreases, and the rate of change of volume can be found using the chain rule: \( \frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt} \).

Front

What is the significance of the first derivative of a function?

Back

The first derivative of a function provides information about the function's rate of change, indicating where the function is increasing or decreasing.

Front

What does it mean for a function to have a local maximum?

Back

A function has a local maximum at a point if the function's value at that point is greater than the values of the function at nearby points.