Textbook Question
Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.4.45c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(c) The average value of a linear function on an interval [a, b] is the function value at the midpoint of [a, b] .
Verified step by step guidance1
Step 1: Recall the formula for the average value of a function f(x) on an interval [a, b]. It is given by: .
Step 2: For a linear function, f(x) = mx + c, substitute this into the formula for the average value. The integral becomes: .
Step 3: Compute the integral of the linear function. The integral of mx is , and the integral of c is cx. Evaluate these terms at the bounds a and b.
Step 4: Simplify the result of the integral and divide by (b - a) to find the average value. After simplification, the average value of the linear function will be: , which is the function value at the midpoint of [a, b].
Step 5: Conclude that the statement is true. The average value of a linear function on an interval [a, b] is indeed equal to the function value at the midpoint of [a, b], as shown by the calculation.
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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 function f(x) over an interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This concept is crucial for understanding how to find the mean value of a function across a specified range, which can differ from simply evaluating the function at a single point.
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Average Value of a Function
Linear Functions
A linear function is a polynomial function of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The properties of linear functions, such as constant slope and direct proportionality, influence how their average value behaves over an interval.
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07:17Linearization
Midpoint of an Interval
The midpoint of an interval [a, b] is calculated as (a + b)/2. For linear functions, evaluating the function at this midpoint can yield the average value, but this is not true for non-linear functions. Understanding this concept helps clarify why the statement may hold true specifically for linear functions.
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07:39Left, Right, & Midpoint Riemann Sums
Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(c) The functions p(𝓍) = sin 3𝓍 and q(𝓍) = 4 sin 3𝓍 are antiderivatives of the same function.
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Textbook Question
Use Table 5.6 to evaluate the following definite integrals.
(c) ∫₃√₂^⁶ d𝓍/(𝓍² ―9)
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Textbook Question
Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(c) Use geometry to find the displacement of the object between t = 2 and t = 5.
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Textbook Question
Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.
(c) Find the mass of the entire rod (0 ≤ x ≤ 10) .
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Textbook Question
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.
∫₁⁷ 1/𝓍 d𝓍 ; n = 6
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