Math Problem Statement

Find the average value of f(x) = 5x + 1/(4x + 3) on [6, 10].

Solution

To find the average value of the function f(x)=5x+14x+3f(x) = 5x + \frac{1}{4x + 3} on the interval [6,10][6, 10], we use the formula for the average value of a function on a closed interval [a,b][a, b]:

Average value of f(x)=1baabf(x)dx\text{Average value of } f(x) = \frac{1}{b - a} \int_a^b f(x) \, dx

Here, a=6a = 6, b=10b = 10, and f(x)=5x+14x+3f(x) = 5x + \frac{1}{4x + 3}.

So, the average value will be:

Average value=1106610(5x+14x+3)dx\text{Average value} = \frac{1}{10 - 6} \int_6^{10} \left( 5x + \frac{1}{4x + 3} \right) dx

Now, let's compute the integral.

  1. The integral of 5x5x is straightforward: 5xdx=5x22\int 5x \, dx = \frac{5x^2}{2}

  2. The integral of 14x+3\frac{1}{4x + 3} is computed using substitution. Let u=4x+3u = 4x + 3, then du=4dxdu = 4 dx, or dx=du4dx = \frac{du}{4}. This gives: 14x+3dx=14ln4x+3\int \frac{1}{4x + 3} dx = \frac{1}{4} \ln|4x + 3|

Thus, the integral becomes:

610(5x+14x+3)dx=[5x22+14ln4x+3]610\int_6^{10} \left( 5x + \frac{1}{4x + 3} \right) dx = \left[ \frac{5x^2}{2} + \frac{1}{4} \ln|4x + 3| \right]_6^{10}

Let me calculate this expression.It seems there was an issue with the calculation. Let me break the steps down and manually handle the evaluation for the integral:

We have the integral to compute:

610(5x+14x+3)dx=[5x22+14ln4x+3]610\int_6^{10} \left( 5x + \frac{1}{4x + 3} \right) dx = \left[ \frac{5x^2}{2} + \frac{1}{4} \ln|4x + 3| \right]_6^{10}

First, compute each part:

  1. For 5x22\frac{5x^2}{2}: 5(10)225(6)22=5×10025×362=25090=160\frac{5(10)^2}{2} - \frac{5(6)^2}{2} = \frac{5 \times 100}{2} - \frac{5 \times 36}{2} = 250 - 90 = 160

  2. For 14ln4x+3\frac{1}{4} \ln|4x + 3|: 14ln4(10)+314ln4(6)+3=14ln(43)14ln(27)\frac{1}{4} \ln|4(10) + 3| - \frac{1}{4} \ln|4(6) + 3| = \frac{1}{4} \ln(43) - \frac{1}{4} \ln(27) Using logarithmic properties: 14ln(4327)\frac{1}{4} \ln \left( \frac{43}{27} \right)

Now, let's add these together and then divide by 106=410 - 6 = 4 to get the average value. Let me calculate this manually.The average value of the function f(x)=5x+14x+3f(x) = 5x + \frac{1}{4x + 3} on the interval [6,10][6, 10] is approximately 40.03.

Would you like a further breakdown of the steps or have any other questions?

Here are some related questions you might explore:

  1. How does changing the interval affect the average value of the function?
  2. How can we compute the average value if f(x)f(x) involves more complex functions like trigonometric or exponential terms?
  3. What is the geometric interpretation of the average value of a function?
  4. Can the average value of a function be negative, and under what conditions?
  5. How does substitution help in simplifying certain integrals?

Tip: When solving definite integrals that involve logarithms, always simplify using logarithmic properties, like lnab=ln(a)ln(b)\ln \frac{a}{b} = \ln(a) - \ln(b).

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Math Problem Analysis

Mathematical Concepts

Average Value of Function
Definite Integrals
Substitution Method
Logarithmic Properties

Formulas

Average value of f(x) = (1 / (b - a)) ∫[a, b] f(x) dx
Integral of 5x is (5x^2 / 2)
Integral of 1 / (4x + 3) is (1/4) ln|4x + 3|

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate Calculus