Question

What is the slope (round your answer to one decimal place)?

Answer 1

Difference quotient formula

`(f(x+h)-f(x))/(h)`

To find the estimation of the slope of f(x) at x = 7, we can use the difference quotient. Substituting with the function f(x) provided, h = 0.001, and x = 7, we get:

`\begin{gathered} f\mleft(x\mright)=9.2x^2-6.4x \\ f(7)=9.2\cdot7^2-6.4\cdot7=406 \\ f\mleft(x+h\mright)=9.2\mleft(x+h\mright)^2-6.4\mleft(x+h\mright) \\ f(7+0.001)=9.2(7.001)^2-6.4(7.001)=406.1224 \\ m=(f(7+0.001)-f(7))/(0.001) \\ m=(406.1224-406)/(0.001) \\ m\approx122.4 \end{gathered}`

Equation of the tangent line

`y-y_1=m(x-x_1)`

The tangent line is tangent to the point (x₁, y₁) which also passes through f(x). And m is the slope of the tangent line.

Given that f(7) = 406, then the tangent point is (7, 406). Using this point and m = 122.4, we get:

`\begin{gathered} y-406=122.4(x-7) \\ y-406=122.4x-122.4(7) \\ y-406=122.4x-856.8 \\ y=122.4x-856.8+406 \\ y=122.4x-450.8\text{ (which is in the mx + b format)} \end{gathered}`