Find the indicated quantities for 

f left parenthesis x right parenthesis equals 2 x squaredf(x)=2x2.

​(A) The slope of the secant line through the points 

​(22​,f(22​)) 

and 

​(22plus+​h,f(22plus+​h)), 

h not equals 0h≠0

​(B) The slope of the graph at 

​(22​,f(22​))

​(C) The equation of the tangent line at 

​(22​,f(22​))

Question content area bottom

Part 1

​(A) The slope of the secant line through the points 

​(22​,f(22​)) 

and 

​(22plus+​h,f(22plus+​h)), 

h not equals 0h≠0​, 

is 

enter your response here.

### (A) The Slope of the Secant Line
The slope of the secant line through the points \((2, f(2))\) and \((2 + h, f(2 + h))\) is given by the formula:

\[
m_{\text{secant}} = \frac{f(2 + h) - f(2)}{(2 + h) - 2}
\]

1. **Find \( f(2) \):**

\[
f(2) = 2(2)^2 = 2 \times 4 = 8
\]

2. **Find \( f(2 + h) \):**

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

3. **Calculate \( m_{\text{secant}} \):**

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

Simplify:

\[
m_{\text{secant}} = \frac{h(8 + 2h)}{h} = 8 + 2h
\]

So, the slope of the secant line is:

\[
m_{\text{secant}} = 8 + 2h
\]

### (B) The Slope of the Graph at \((2, f(2))\)
The slope of the graph at a point is given by the derivative of \(f(x)\) evaluated at that point.

1. **Find \( f'(x) \):**

\[
f'(x) = \frac{d}{dx}(2x^2) = 4x
\]

2. **Evaluate \( f'(2) \):**

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

So, the slope of the graph at \((2, f(2))\) is \(8\).

### (C) The Equation of the Tangent Line at \((2, f(2))\)
The equation of the tangent line is given by:

\[
y - f(2) = f'(2)(x - 2)
\]

1. We know:
   - \(f(2) = 8\)
   - \(f'(2) = 8\)

2. Substitute these values:

\[
y - 8 = 8(x - 2)
\]

3. Simplify:

\[
y - 8 = 8x - 16 \quad \Rightarrow \quad y = 8x - 8
\]

So, the equation of the tangent line is:

\[
y = 8x - 8
\]

Would you like any further details or have any additional questions?

### Follow-up Questions:
1. How do we interpret the meaning of the secant line in relation to the curve \(f(x)\)?
2. What is the geometric significance of the slope of the tangent line?
3. How can we find the equation of a normal line to \(f(x)\) at \(x = 2\)?
4. What happens to the secant line slope as \(h\) approaches zero?
5. Can we find the second derivative \(f''(x)\) and interpret its meaning at \(x = 2\)?

### Tip:
For finding tangent lines, always remember that the slope of the tangent line is the value of the derivative of the function at the given point.

Find the indicated quantities for f(x) = 2x^2. (A) The slope of the secant line through the points (2, f(2)) and (2 + h, f(2 + h)), h ≠ 0 (B) The slope of the graph at (2, f(2)) (C) The equation of the tangent line at (2, f(2))

This problem involves finding the slope of a secant line through (2, f(2)) and (2 + h, f(2 + h)), the slope of the graph at (2, f(2)), and the equation of the tangent line for f(x) = 2x^2. It introduces key calculus concepts such as derivatives and secant line slopes. Suitable for Grades 11-12 students.

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