Math Problem Statement

Let f be the function defined by f(x) = x + ln x. What is the value of c for which the instantaneous rate of change of f at x = c is the same as the average rate of change of f over [1,4]?

Solution

To solve this problem, we need to find the value of cc where the instantaneous rate of change of f(x)=x+lnxf(x) = x + \ln x at x=cx = c is equal to the average rate of change of ff over the interval [1,4][1, 4].

Step 1: Find the Average Rate of Change of ff over [1,4][1, 4]

The average rate of change of f(x)f(x) over [a,b][a, b] is given by: f(b)f(a)ba\frac{f(b) - f(a)}{b - a} For this problem, a=1a = 1 and b=4b = 4.

Calculate f(1)f(1) and f(4)f(4):

  • f(1)=1+ln(1)=1+0=1f(1) = 1 + \ln(1) = 1 + 0 = 1
  • f(4)=4+ln(4)=4+ln(4)4+1.386=5.386f(4) = 4 + \ln(4) = 4 + \ln(4) \approx 4 + 1.386 = 5.386

Now, the average rate of change is: f(4)f(1)41=5.38613=4.38631.462\frac{f(4) - f(1)}{4 - 1} = \frac{5.386 - 1}{3} = \frac{4.386}{3} \approx 1.462

Step 2: Find the Instantaneous Rate of Change of ff at x=cx = c

The instantaneous rate of change of f(x)=x+lnxf(x) = x + \ln x is the derivative f(x)f'(x).

  1. Differentiate f(x)f(x): f(x)=1+1xf'(x) = 1 + \frac{1}{x}

  2. Set f(c)f'(c) equal to the average rate of change and solve for cc: 1+1c=1.4621 + \frac{1}{c} = 1.462 1c=1.4621=0.462\frac{1}{c} = 1.462 - 1 = 0.462 c=10.4622.164c = \frac{1}{0.462} \approx 2.164

Answer

The value of cc is approximately 2.1642.164, which corresponds to option C.

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions for practice:

  1. How would you find the instantaneous rate of change at any given point?
  2. What does the derivative of f(x)=x+lnxf(x) = x + \ln x represent in this context?
  3. How would the process change if we were given a different function, such as f(x)=x2+lnxf(x) = x^2 + \ln x?
  4. What if the interval was [1,3][1, 3] instead of [1,4][1, 4]? How would that affect the result?
  5. How would you interpret the result of the average rate of change in real-world terms?

Tip: Always make sure to carefully evaluate logarithmic functions to avoid approximation errors.

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Math Problem Analysis

Mathematical Concepts

Calculus
Rates of Change
Differentiation
Logarithmic Functions

Formulas

Average rate of change: (f(b) - f(a)) / (b - a)
Instantaneous rate of change (derivative): f'(x)

Theorems

Mean Value Theorem

Suitable Grade Level

Grades 11-12 or College Calculus