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Chapter 4: Problem 3

Find the average value of \(f(x)=6 x+5\) over [2,6] average value = _____________

Short Answer

Expert verified
The average value is 29.

Step by step solution

01

Identify the formula for the average value

The average value of a function over the interval \([a, b]\) is given by \(\frac{1}{b-a} \int_a^b f(x) \, dx\). For this problem, \[a = 2\text{ and }b = 6\text{ and }f(x) = 6x + 5.\]
02

Set up the integral

Set up the definite integral \(\frac{1}{6-2} \int_2^6 (6x + 5) \, dx\). This simplifies to \(\frac{1}{4} \int_2^6 (6x + 5) \, dx\).
03

Find the antiderivative

Compute the antiderivative of \(6x + 5\). The antiderivative is \(3x^2 + 5x\).
04

Evaluate the definite integral

Evaluate the antiderivative from \2 \text{ to } 6\: \[: 3x^2 + 5x\]_2^6\. Substituting in the bounds, we get\[ (3(6^2) + 5(6)) - (3(2^2) + 5(2))\].
05

Calculate the result

Perform the arithmetic: \( (3(36) + 30) - (3(4) + 10) = (108 + 30) - (12 + 10) = 138 - 22 = 116\).
06

Divide by the interval length

Divide by the length of the interval \([a, b]\): \( \frac{116}{4} = 29\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

definite integral
The definite integral is a fundamental concept in calculus that represents the area under a curve between two points on the x-axis. For a given function, the definite integral from point \( a \) to point \( b \) is denoted as \( \int_a^b f(x) \, dx \). This integral is computed by evaluating the area under the curve of \( f(x) \) between the limits \( a \) and \( b \). It's important to understand that the definite integral provides a numerical value representing this area. For our exercise, the definite integral we set up was \( \int_2^6 (6x + 5) \, dx \), which helps us in finding the average value of the given function over the specified interval.
antiderivative
An antiderivative, also known as an indefinite integral, is the reverse process of differentiation. It helps in finding the original function from its derivative. In the context of definite integrals, the antiderivative is used to find the area under the curve. For instance, to find the average value of the function \( f(x) = 6x + 5 \), we first need its antiderivative. For \(6x + 5\), the antiderivative is \( 3x^2 + 5x \). This is obtained by reversing the differentiation process, where we integrate each term of the given function separately.
interval length division
To find the average value of a function over an interval, we use the definite integral and then divide by the length of the interval. This step ensures we normalize the total area under the curve by the width of the interval, providing the mean value over that span. Mathematically, this is expressed as: \( \text{{Average value}} = \frac{1}{b-a} \int_a^b f(x) \, dx \). For our exercise:
  • The interval is \([2, 6]\), thus \( b - a = 6 - 2 = 4 \).
  • After evaluating the integral, we divide the resulting area by 4. This calculation yields the average value of the function \( 6x + 5 \) over the interval \([2, 6]\).
evaluation of definite integrals
Once we have computed the antiderivative of our function, the next step is to evaluate it within the bounds of the interval. This involves substituting the upper and lower limits into the antiderivative and subtracting the latter from the former. For our function \( 6x + 5 \):
  • The antiderivative is \( 3x^2 + 5x \).
  • We evaluate it from 2 to 6, substituting these values gives us \( (3(6^2) + 5(6)) - (3(2^2) + 5(2)) \).
  • This computation simplifies to \( (108 + 30) - (12 + 10) \), resulting in \( 138 - 22 = 116 \). Finally, we divide by the interval length (4) to find the average value: \( \frac{116}{4} = 29 \).

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Most popular questions from this chapter

Filters at a water treatment plant become dirtier over time and thus become less effective; they are replaced every 30 days. During one 30 -day period, the rate at which pollution passes through the filters into a nearby lake (in units of particulate matter per day) is measured every 6 days and is given in the following table. The time \(t\) is measured in days since the filters were replaced. $$ \begin{array}{lcccccc} \hline \text { Day, } t & 0 & 6 & 12 & 18 & 24 & 30 \\ \hline \text { Rate of pollution in units per day, } p(t) & 7 & 8 & 10 & 13 & 18 & 35 \\ \hline \end{array} $$ Table 4.1.12: Pollution data for the water filters. a. Plot the given data on a set of axes with time on the horizontal axis and the rate of pollution on the vertical axis. b. Explain why the amount of pollution that entered the lake during this 30 -day period would be given exactly by the area bounded by \(y=p(t)\) and the \(t\) -axis on the time interval [0,30] . c. Estimate the total amount of pollution entering the lake during this 30 -day period. Carefully explain how you determined your estimate.

On a sketch of \(y=e^{x},\) represent the left Riemann sum with \(n=2\) approximating \(\int_{2}^{3} e^{x} d x\). Write out the terms of the sum, but do not evaluate it: Sum \(=\) ___________________ + ___________________ On another sketch, represent the right Riemann sum with \(n=2\) approximating \(\int_{2}^{3} e^{x} d x\). Write out the terms of the sum, but do not evaluate it: Sum \(=\) ___________________ + ___________________ Which sum is an overestimate? Which sum is an underestimate?

A toy rocket is launched vertically from the ground on a day with no wind. The rocket's vertical velocity at time \(t\) (in seconds) is given by \(v(t)=500-32 t\) feet \(/ \mathrm{sec}\). a. At what time after the rocket is launched does the rocket's velocity equal zero? Call this time value \(a\). What happens to the rocket at \(t=a\) ? b. Find the value of the total area enclosed by \(y=v(t)\) and the \(t\) -axis on the interval \(0 \leq t \leq a\). What does this area represent in terms of the physical setting of the problem? c. Find an antiderivative \(s\) of the function \(v\). That is, find a function \(s\) such that \(s^{\prime}(t)=v(t)\). d. Compute the value of \(s(a)-s(0) .\) What does this number represent in terms of the physical setting of the problem? e. Compute \(s(5)-s(1)\). What does this number tell you about the rocket's flight?

A function \(f\) is given piecewise by the formula $$ f(x)=\left\\{\begin{array}{ll} -x^{2}+2 x+1, & \text { if } 0 \leq x<2 \\ -x+3, & \text { if } 2 \leq x<3 \\ x^{2}-8 x+15, & \text { if } 3 \leq x \leq 5 \end{array}\right. $$ a. Determine the exact value of the net signed area enclosed by \(f\) and the \(x\) -axis on the interval [2,5] b. Compute the exact average value of \(f\) on [0,5] . c. Find a formula for a function \(g\) on \(5 \leq x \leq 7\) so that if we extend the above definition of \(f\) so that \(f(x)=g(x)\) if \(5 \leq x \leq 7,\) it follows that \(\int_{0}^{7} f(x) d x=0\)

Suppose that the velocity of a moving object is given by \(v(t)=t(t-1)(t-3),\) measured in feet per second, and that this function is valid for \(0 \leq t \leq 4\) a. Write an expression involving definite integrals whose value is the total change in position of the object on the interval [0,4] . b. Use appropriate technology (such as http://gvsu.edu/s/a93) to compute Riemann sums to estimate the object's total change in position on [0,4] . Work to ensure that your estimate is accurate to two decimal places, and explain how you know this to be the case. c. Write an expression involving definite integrals whose value is the total distance traveled by the object on [0,4] d. Use appropriate technology to compute Riemann sums to estimate the object's total distance travelled on [0,4] . Work to ensure that your estimate is accurate to two decimal places, and explain how you know this to be the case. e. What is the object's average velocity on [0,4] , accurate to two decimal places?
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