use the limit definition to find the expression 2x squared minus 7x plus 5 for the slope of the tangent line at any given point (x,f(x)). Your answer is in terms of x

To find the slope of the tangent line to the curve \( f(x) = 2x^2 - 7x + 5 \) at any given point \((x, f(x))\), we use the limit definition of the derivative.

The slope of the tangent line at a point \( x \) is given by the derivative \( f'(x) \), which is defined as:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

1. **Step 1: Calculate \( f(x+h) \)**

Given \( f(x) = 2x^2 - 7x + 5 \), we need to find \( f(x+h) \):

\[
f(x+h) = 2(x+h)^2 - 7(x+h) + 5
\]

Expanding \( f(x+h) \):

\[
f(x+h) = 2(x^2 + 2xh + h^2) - 7x - 7h + 5
\]

\[
= 2x^2 + 4xh + 2h^2 - 7x - 7h + 5
\]

2. **Step 2: Find \( f(x+h) - f(x) \)**

Now, subtract \( f(x) = 2x^2 - 7x + 5 \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 - 7x - 7h + 5) - (2x^2 - 7x + 5)
\]

\[
= 2x^2 + 4xh + 2h^2 - 7x - 7h + 5 - 2x^2 + 7x - 5
\]

\[
= 4xh + 2h^2 - 7h
\]

3. **Step 3: Divide by \( h \) and take the limit**

\[
f'(x) = \lim_{h \to 0} \frac{4xh + 2h^2 - 7h}{h}
\]

\[
= \lim_{h \to 0} (4x + 2h - 7)
\]

4. **Step 4: Evaluate the limit as \( h \to 0 \)**

\[
f'(x) = 4x - 7
\]

So, the slope of the tangent line to the curve \( f(x) = 2x^2 - 7x + 5 \) at any given point \( (x, f(x)) \) is \( 4x - 7 \).

### Summary
The expression for the slope of the tangent line at any point on the curve \( f(x) = 2x^2 - 7x + 5 \) is:

\[
f'(x) = 4x - 7
\]

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

### Related Questions:
1. How would you find the tangent line at a specific point, say \( x = 3 \), on the curve \( f(x) = 2x^2 - 7x + 5 \)?
2. Can you use this method to find the derivative of other polynomial functions, such as \( f(x) = 3x^3 - 4x^2 + x - 2 \)?
3. What is the geometric interpretation of the derivative \( f'(x) = 4x - 7 \)?
4. How would you find the points where the tangent line is horizontal on the curve \( f(x) = 2x^2 - 7x + 5 \)?
5. How can the derivative be used to find the local minima and maxima of the function \( f(x) = 2x^2 - 7x + 5 \)?

### Tip:
When working with derivatives, always double-check each algebraic step to ensure the correctness of the expressions you derive, especially when expanding and simplifying polynomials.

Learn how to calculate the slope of the tangent line to the curve f(x) = 2x^2 - 7x + 5 at any point (x, f(x)) using the limit definition of the derivative. Detailed step-by-step solution provided.

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