I need help on this question, If you could help that would be awesome

Question

Find the difference quotient of​ f; that​ is, find f(x+h)-f(x)/h (all of this is over h not just f(x)), h doesnt not = 0

f(x) =x^2-5x+7

Answer 1

this is difficult without math equations, but here goes:

m (the slope or derivative) = lim h approaches 0 of change in y divided by change in x

change in y = F(x+h) - F(x) change in x = h h approaching 0

for Y = f(x) = x^2 -5x +7

F(x+h) = (x+h)^2 - 5(x+h) + 7 all over h

F(x+h) = x^2 +2xh + h^2 -5x -5h +7 all over h

F(x+h) - f(x) = (x^2 +2xh + h^2 -5x -5h +7) - (x^2 - 5x +7) all over h

simplify = 2xh + h^2 -5h all over h

= 2x +h -5 (cancel the denominator)

lim h approaching 0 for 2x + h - 5 = 2x -5

m (the slope) f(x) = x^2 -5x +7 is equal to 2x-5.

This is much easier to see written better. Feel free to message me, I'll show you on a white board, no charge.

Answer 2

To calculate the difference quotient [f(x + h) − f(x)] / h, first find expressions for f(x) and f(x + h). The expression for f(x) = x² − 5x + 7 is given, but it's necessary to calculate f(x + h) by substituting x + h wherever x appears in the function.

f(x) = x² − 5x + 7

substitute x + h for x to calculate f(x + h)

f(x + h) = (x + h)² − 5(x + h) + 7

f(x + h) = (x² + 2xh + h²) − 5x − 5h + 7

Now plug these expressions for f(x + h) and f(x) into [f(x + h) − f(x)] / h to calculate the difference quotient.

[f(x + h) − f(x)] / h

[(x² + 2xh + h² − 5x − 5h + 7) − (x² − 5x + 7)] / h

x², 5x, and 7 cancel out

[2xh − 5h + h²] / h

Divide each term by h

2x − 5 + h

The difference quotient is 2x − 5 + h.

Note: The difference quotient [f(x + h) − f(x)] / h represents the slope of the line segment between points on the graph of f, with x-values x and x + h (a value h units away) and corresponding y-values f(x) and f(x + h).