Unit 4-5 Test Review Practice Problems

Front

What is average acceleration?

Back

Average acceleration is defined as the change in velocity divided by the time taken for that change. It can be calculated using the formula: \( a_{avg} = \frac{\Delta v}{\Delta t} \).

Front

How do you find the average rate of change of a function over an interval?

Back

The average rate of change of a function \( f(x) \) over the interval \( [a, b] \) is calculated using the formula: \( \frac{f(b) - f(a)}{b - a} \).

Front

What does the Intermediate Value Theorem (IVT) state?

Back

The Intermediate Value Theorem states that if a function is continuous on the interval \( [a, b] \), then it takes every value between \( f(a) \) and \( f(b) \) at least once.

Front

What is concavity in relation to a function?

Back

A function is concave up on an interval if its second derivative is positive on that interval, indicating that the graph of the function is curving upwards.

Front

How do you determine intervals of concavity from a derivative graph?

Back

To determine intervals of concavity from a derivative graph, look for where the first derivative is increasing (concave up) or decreasing (concave down).

Front

What is linear approximation?

Back

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

Front

What is the formula for average acceleration over the interval \( [2, 5] \)?

Back

Average acceleration can be calculated as: \( a_{avg} = \frac{v(5) - v(2)}{5 - 2} \).

Front

How do you apply the IVT to find guaranteed values of a function?

Back

To apply the IVT, check if the function is continuous on the interval and then find values between \( f(a) \) and \( f(b) \) that the function must take.

Front

What is the significance of the value \( f(c) = 4 \) in the context of the IVT?

Back

The value \( f(c) = 4 \) is significant because it is a value that the function must attain between the endpoints of the interval due to the continuity of the function.

Front

What does it mean for a function to be increasing?

Back

A function is increasing on an interval if, for any two points \( x_1 \) and \( x_2 \) in that interval where \( x_1 < x_2 \), it holds that \( f(x_1) < f(x_2) \).