Math Problem Statement

Find the value of f(x+h) - f(x) /H, if f(x)=x2+7 $$$$

Solution

To find the value of f(x+h)f(x)h\frac{f(x+h) - f(x)}{h} for the function f(x)=x2+7f(x) = x^2 + 7, we need to carefully evaluate each step. Here's the detailed process:

  1. Evaluate f(x+h)f(x + h):

    Given f(x)=x2+7f(x) = x^2 + 7, substitute x+hx + h into the function: f(x+h)=(x+h)2+7f(x + h) = (x + h)^2 + 7

  2. Expand f(x+h)f(x + h):

    f(x+h)=(x+h)2+7=x2+2xh+h2+7f(x + h) = (x + h)^2 + 7 = x^2 + 2xh + h^2 + 7

  3. Find f(x+h)f(x)f(x + h) - f(x):

    Now, compute the difference: f(x+h)f(x)=(x2+2xh+h2+7)(x2+7)f(x + h) - f(x) = (x^2 + 2xh + h^2 + 7) - (x^2 + 7) Simplify the expression: f(x+h)f(x)=x2+2xh+h2+7x27=2xh+h2f(x + h) - f(x) = x^2 + 2xh + h^2 + 7 - x^2 - 7 = 2xh + h^2

  4. Divide by hh:

    f(x+h)f(x)h=2xh+h2h\frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2}{h} Simplify the fraction: f(x+h)f(x)h=2x+h\frac{f(x + h) - f(x)}{h} = 2x + h

So, the value of f(x+h)f(x)h\frac{f(x + h) - f(x)}{h} is 2x+h2x + h.

If you need further clarification or have any questions, feel free to ask!

Here are some related questions to explore:

  1. What is the derivative of f(x)=x2+7f(x) = x^2 + 7 using the limit definition of a derivative?
  2. How does the expression f(x+h)f(x)h\frac{f(x + h) - f(x)}{h} help in understanding the concept of a secant line?
  3. What happens to f(x+h)f(x)h\frac{f(x + h) - f(x)}{h} as hh approaches 0?
  4. Can you generalize the process of finding f(x+h)f(x)h\frac{f(x + h) - f(x)}{h} for any polynomial function?
  5. How would you interpret the result 2x+h2x + h in the context of the rate of change?
  6. What are the implications of this result for the function's behavior near xx?
  7. Can you find f(x+h)f(x)h\frac{f(x + h) - f(x)}{h} for the function f(x)=sin(x)f(x) = \sin(x)?
  8. How does the concept of a limit apply to finding derivatives?

Tip: Always double-check your algebraic manipulations, especially when expanding expressions and simplifying fractions. This attention to detail ensures accuracy in your calculations.

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Math Problem Analysis

Mathematical Concepts

Functions
Quadratic Equations
Limits

Formulas

Quadratic function formula: f(x) = x^2 + 7
Limit definition of a derivative

Theorems

Definition of a derivative

Suitable Grade Level

Grades 11-12