HW-Average Value of a Function

Front

What is the average value of a function over an interval [a, b]?

Back

The average value of a function f(x) over the interval [a, b] is given by the formula: \( \frac{1}{b-a} \int_a^b f(x) \, dx \).

Front

How do you find the integral of a function?

Back

To find the integral of a function f(x), you calculate \( \int f(x) \, dx \), which represents the area under the curve of f(x) from a specified lower limit to an upper limit.

Front

What is the significance of the limits in an integral?

Back

The limits of an integral define the interval over which the function is being integrated. The lower limit is the starting point, and the upper limit is the endpoint of the interval.

Front

What does the notation \( \int_a^b f(x) \, dx \) represent?

Back

This notation represents the definite integral of the function f(x) from x = a to x = b, which calculates the net area between the function and the x-axis over that interval.

Front

What is the geometric interpretation of the definite integral?

Back

The definite integral represents the net area between the graph of the function and the x-axis over the interval [a, b]. Areas above the x-axis are positive, while areas below are negative.

Front

How do you determine which integral has the largest value?

Back

To determine which integral has the largest value, evaluate each integral over the specified limits and compare the results.

Front

What is the formula for the area under a linear function?

Back

For a linear function f(x) = mx + b, the area under the curve from x = a to x = b can be calculated using the formula: \( \text{Area} = \frac{1}{2} \times (b-a) \times (f(a) + f(b)) \).

Front

What is the relationship between the average value of a function and its integral?

Back

The average value of a function over an interval is directly related to its integral, as it is calculated by dividing the integral of the function over that interval by the length of the interval.

Front

What is the Fundamental Theorem of Calculus?

Back

The Fundamental Theorem of Calculus states that if F is an antiderivative of f on an interval [a, b], then \( \int_a^b f(x) \, dx = F(b) - F(a) \).

Front

How do you evaluate the integral \( \int_0^4 (4-x) \, dx \)?

Back

To evaluate \( \int_0^4 (4-x) \, dx \), find the antiderivative, which is \( 4x - \frac{x^2}{2} \), and then compute \( F(4) - F(0) \). The result is 8.