Question

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. (Simplify your answers completely.) f(x) = 8x^2 + 9x

Step 1: f(x + h) = ______
Step 2: f(x + h) − f(x) = _____
Step 3: f(x + h) − f(x) h = _____
Step 4: f '(x) = lim h→0 f(x + h) − f(x) h =____

Answer 1

The slope of the tangent line to the function f(x) = 8x^2 + 9x at any point is found by taking the derivative, which is 16x + 9.

Step-by-step explanation:

To find the slope of the tangent line to the function f(x) = 8x^2 + 9x at any point using the four-step process, follow these steps:

1. Calculate f(x + h) by substituting x + h into the function: f(x + h) = 8(x + h)^2 + 9(x + h).
2. Subtract f(x) from f(x + h) to find the difference: f(x + h) − f(x) = [8(x + h)^2 + 9(x + h)] - [8x^2 + 9x].
3. Divide the difference by h to find the average rate of change.
4. Take the limit as h approaches 0 to find the derivative, which is the slope of the tangent line: f '(x) = lim h→0 = 16x + 9.

Therefore, the slope of the tangent line to the given function at any point is 16x + 9, which is the derivative of the function.