Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Average Value of a Function

Calculus

8. Definite Integrals

Average Value of a Function

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3:06 minutes

Problem 5.4.53a

Textbook Question

Average value with a parameter Consider the function ƒ(𝓍) = a𝓍 (1―𝓍) on the interval [0, 1], where a is a positive real number.
(a) Find the average value of ƒ as a function of a .

Verified step by step guidance

Step 1: Recall the formula for the average value of a function ƒ(𝓍) on the interval [a, b], which is given by:

\frac{1}{(b - a)} \int f_{x} d x

. In this case, the interval is [0, 1] and the function is ƒ(𝓍) = a𝓍(1 - 𝓍).

Step 2: Substitute the interval [0, 1] and the function ƒ(𝓍) = a𝓍(1 - 𝓍) into the formula for average value:

\frac{1}{1} \int 0^1 (a x (1 - x)) d x

. This simplifies to:

\int 0^1 a x (1 - x) d x

Step 3: Expand the integrand a𝓍(1 - 𝓍) to simplify the integral. This becomes:

a (x - x^{2})

. The integral now looks like:

a \int 0^1 (x - x^{2}) d x

Step 4: Break the integral into two separate parts:

a (\int 0^1 x d x - \int 0^1 x^{2} d x)

. Compute each integral separately:

\int 0^1 x d x

and

\int 0^1 x^{2} d x

. Use the power rule for integration:

\int x^{n} d x = \frac{x^{n + 1}}{n + 1}

Step 5: After computing the integrals, combine the results and multiply by the constant 'a' to find the average value of ƒ(𝓍) as a function of 'a'. The final expression will represent the average value of the function over the interval [0, 1].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Value of a Function

The average value of a continuous function ƒ over an interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] ƒ(x) dx. This concept is essential for determining how the function behaves on the specified interval, providing a single representative value that summarizes the function's overall trend.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval. In this context, it is used to compute the integral of the function ƒ(x) = a𝓍(1 - 𝓍) from 0 to 1, which is necessary for finding the average value of the function.

Parameter in Functions

A parameter is a variable that influences the behavior of a function but is not the primary variable of interest. In this case, 'a' is a parameter that affects the shape and scale of the function ƒ(x) = a𝓍(1 - 𝓍), and understanding its role is crucial for expressing the average value as a function of 'a'.

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