Suppose f(x) = 2x^2+C
, where C
 is any real number. Then the expression

 (f(2+h)-f(2))/h

can be written in the form Ah+B(2)
, where A 
 and B
 are constants. Find:

(a) 
 
A=


(b) 
 
B=

(c) 
 
f'(2)

We are given the function \( f(x) = 2x^2 + C \), where \( C \) is a constant, and we are tasked with finding the limit of the difference quotient:

\[
\frac{f(2+h) - f(2)}{h}
\]

### Step 1: Calculate \( f(2+h) \)

Using the given function \( f(x) = 2x^2 + C \), we calculate \( f(2+h) \):

\[
f(2+h) = 2(2+h)^2 + C = 2(4 + 4h + h^2) + C = 8 + 8h + 2h^2 + C
\]

### Step 2: Calculate \( f(2) \)

Next, we calculate \( f(2) \):

\[
f(2) = 2(2^2) + C = 8 + C
\]

### Step 3: Difference of \( f(2+h) \) and \( f(2) \)

Now, subtract \( f(2) \) from \( f(2+h) \):

\[
f(2+h) - f(2) = (8 + 8h + 2h^2 + C) - (8 + C) = 8h + 2h^2
\]

### Step 4: Expression \( \frac{f(2+h) - f(2)}{h} \)

Now, divide by \( h \):

\[
\frac{f(2+h) - f(2)}{h} = \frac{8h + 2h^2}{h} = 8 + 2h
\]

### Step 5: Express in the form \( Ah + B \)

The expression \( 8 + 2h \) is already in the form \( Ah + B \), where \( A = 2 \) and \( B = 8 \).

Thus:
- (a) \( A = 2 \)
- (b) \( B = 8 \)

### Step 6: Find \( f'(2) \)

The derivative \( f'(x) \) of the function \( f(x) = 2x^2 + C \) is:

\[
f'(x) = 4x
\]

Therefore, \( f'(2) \) is:

\[
f'(2) = 4(2) = 8
\]

Thus, the answers are:

- (a) \( A = 2 \)
- (b) \( B = 8 \)
- (c) \( f'(2) = 8 \)

---

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

Here are 5 related questions to extend your understanding:
1. How do you find the derivative of any quadratic function?
2. What is the difference quotient and why is it useful in calculus?
3. How does the constant \( C \) affect the derivative of the function?
4. Can you find \( f'(x) \) for other values of \( x \) in this problem?
5. How would the difference quotient change if the function was cubic?

**Tip:** The difference quotient is a key concept in calculus that leads to the derivative, which measures the rate of change of a function at a point.

Suppose f(x) = 2x^2 + C, where C is any real number. Then the expression (f(2+h)-f(2))/h can be written in the form Ah+B, where A and B are constants. Find: (a) A = (b) B = (c) f'(2) =

Explore how to find the difference quotient of the function f(x) = 2x^2 + C and its derivative at x = 2. This problem includes concepts of calculus, such as derivatives and difference quotients, with a step-by-step breakdown. Suitable for students in Grades 11-12 or early university.

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