Math Problem Statement

Given f(x) = 1/x+11, find the average rate of change of f (x) on the interval [3,3 + h] your answer will be a expression involving h

Solution

To find the average rate of change of the function f(x)=1x+11f(x) = \frac{1}{x} + 11 on the interval [3,3+h][3, 3 + h], we use the formula for the average rate of change:

Average Rate of Change=f(3+h)f(3)(3+h)3\text{Average Rate of Change} = \frac{f(3 + h) - f(3)}{(3 + h) - 3}

Step 1: Find f(3+h)f(3 + h)

Substitute 3+h3 + h into the function:

f(3+h)=13+h+11f(3 + h) = \frac{1}{3 + h} + 11

Step 2: Find f(3)f(3)

Substitute 33 into the function:

f(3)=13+11f(3) = \frac{1}{3} + 11

Step 3: Calculate the difference in function values

Now subtract f(3)f(3) from f(3+h)f(3 + h):

f(3+h)f(3)=(13+h+11)(13+11)f(3 + h) - f(3) = \left( \frac{1}{3 + h} + 11 \right) - \left( \frac{1}{3} + 11 \right) The 11's cancel out, so you're left with:

f(3+h)f(3)=13+h13f(3 + h) - f(3) = \frac{1}{3 + h} - \frac{1}{3}

Step 4: Simplify the difference

To simplify 13+h13\frac{1}{3 + h} - \frac{1}{3}, we combine the fractions:

13+h13=3(3+h)3(3+h)=h3(3+h)\frac{1}{3 + h} - \frac{1}{3} = \frac{3 - (3 + h)}{3(3 + h)} = \frac{-h}{3(3 + h)}

Step 5: Divide by hh

Now, divide the result by hh to find the average rate of change:

Average Rate of Change=h3(3+h)h=13(3+h)\text{Average Rate of Change} = \frac{\frac{-h}{3(3 + h)}}{h} = \frac{-1}{3(3 + h)}

Final Answer:

The average rate of change of f(x)f(x) on the interval [3,3+h][3, 3 + h] is:

13(3+h)\boxed{\frac{-1}{3(3 + h)}}

Would you like further details or clarifications?


Related Questions:

  1. How do you find the instantaneous rate of change of f(x)f(x) at x=3x = 3?
  2. What is the derivative of f(x)=1x+11f(x) = \frac{1}{x} + 11?
  3. Can you generalize the average rate of change for any interval [a,a+h][a, a+h]?
  4. What happens to the average rate of change as h0h \to 0?
  5. How would the result change if the function was f(x)=2x+11f(x) = \frac{2}{x} + 11?

Tip: The average rate of change is closely related to the concept of the derivative, which gives the instantaneous rate of change at a specific point.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Average Rate of Change

Formulas

Average Rate of Change = (f(b) - f(a)) / (b - a)
f(x) = 1/x + 11

Theorems

Difference Quotient (used in rate of change)

Suitable Grade Level

Grades 10-12