The limit represents f ′(c) for a function f(x) and a number c. Find f(x) and c.
lim 
x→25 
8

x
 − 40
x − 25
f(x)	=

To find the function \( f(x) \) and the number \( c \) for which the given limit represents the derivative \( f'(c) \), we start by analyzing the limit expression:

\[
\lim_{x \to 25} \frac{8\sqrt{x} - 40}{x - 25}
\]

This limit has the form of a derivative:

\[
f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}
\]

### Step 1: Identify \( c \)

The expression inside the limit, \( x \to 25 \), indicates that \( c = 25 \).

### Step 2: Identify \( f(x) \)

To match the derivative form, we compare:

\[
\frac{8\sqrt{x} - 40}{x - 25}
\]

This can be written as:

\[
\frac{f(x) - f(25)}{x - 25}
\]

By comparing the numerator \( 8\sqrt{x} - 40 \) with \( f(x) - f(25) \), we identify:

\[
f(x) = 8\sqrt{x}
\]

Now, let's compute \( f(25) \):

\[
f(25) = 8\sqrt{25} = 8 \times 5 = 40
\]

Thus, the expression:

\[
\frac{8\sqrt{x} - 40}{x - 25}
\]

is indeed in the form of the derivative \( \frac{f(x) - f(25)}{x - 25} \), where \( f(x) = 8\sqrt{x} \) and \( c = 25 \).

### Step 3: Conclusion

- \( f(x) = 8\sqrt{x} \)
- \( c = 25 \)

This confirms that the limit represents the derivative \( f'(c) \) for \( f(x) = 8\sqrt{x} \) at \( c = 25 \).

Would you like further details on this solution or have any questions?

### Related Questions

1. How do you compute derivatives using the definition of the derivative?
2. What are the properties of square root functions?
3. How do you interpret the meaning of the derivative in a real-world context?
4. How do you find the tangent line to a curve at a specific point?
5. How does the chain rule apply to functions involving square roots?

**Tip:** When finding derivatives, always check if the limit matches the form of the derivative's definition!

The limit represents f ′(c) for a function f(x) and a number c. Find f(x) and c. lim x→25 (8√x − 40)/(x − 25)

Understand how to find the derivative of the square root function f(x) = 8√x at c = 25 using the limit definition. This solution covers key calculus concepts, including limits and the definition of the derivative, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation