Question

Find the difference quotient g(x)=5x^2+7x-10

Answer 1

The difference quotient of a function f(x) is found by evaluating the function at two points, x and x+h, and dividing the difference in their function values by h.

For the function g(x) = 5x^2 + 7x - 10, we can find the difference quotient. Let's evaluate g(x) at x and x+h:

g(x) = 5x^2 + 7x - 10
g(x+h) = 5(x+h)^2 + 7(x+h) - 10

Expanding both expressions:

g(x) = 5x^2 + 7x - 10
g(x+h) = 5(x^2 + 2xh + h^2) + 7x + 7h - 10

Now, subtracting g(x) from g(x+h):

g(x+h) - g(x) = 5(x^2 + 2xh + h^2) + 7x + 7h - 10 - (5x^2 + 7x - 10)
= 5x^2 + 10xh + 5h^2 + 7x + 7h - 10 - 5x^2 - 7x + 10
= 10xh + 5h^2 + 7h

Finally, dividing by h to get the difference quotient:

(g(x+h) - g(x))/h = (10xh + 5h^2 + 7h)/h
= 10x + 5h + 7

Therefore, the difference quotient of g(x) = 5x^2 + 7x - 10 is 10x + 5h + 7.