Could someone help me with this algebra question please

Question

Find (f(x+h) - f(x))/h if f(x) = 5x^2 + 4

Answer 1

This, while an algebra problem, is a formula you usually see at the beginning of calculus classes; as h goes to 0, the formula approaches the slope of the function at x (also known as the first derivative). Note that, while the denominator is written as h, it could be written as x + h - h, which tells us that the formula is the slope of a line from (x, f(x) ) to (x+h, f(x+h))

f(x+h) = 5 * (x+h)^2 + 4 = 5(x^2 + 2hx + h^2) + 4 = 5x^2 + 10hx + 5h^2 + 4

f(x) = 5x^2 + 4

Thus, f(x+h) - f(x) = 5x^2 + 10hx + 5h^2 + 4 - (5x^2 + 4) = 10hx + 5h^2, and

(f(x+h)-f(x))/h = (10hx + 5h^2)/h = 10x + 5h

As h goes to 0, this goes to 10x, which tells us that the slope of the function at x is 10x.

Answer 2

Question: Find (f(x+h) - f(x))/h if f(x) = 5x^2 + 4

Answer: This is how they introduce you to calculus very early on in life. Let's do this!

So you need to do some expanding in order to get this problem going. The notation f(x+h) means that wherever I see x in the function, I replace it with x+h. Here's what that looks like:

f(x+h) = 5(x + h)² + 4.

f(x) = 5x² + 4. (This has not changed.)

So let's expand f(x+h). Remember how to expand a binomial?

Recall: (a + b)² = a² + 2ab + b². This formula comes from using the FOIL method: (a + b)(a + b).

(x + h)² = x² + 2xh + h². Multiply 5 to this: 5x² + 10xh + 5h².

Add 4: 5x² + 10xh + 5h² + 4. This is how f(x + h) is expanded. Now let's incorporate it into a large fraction:

[5x² + 10xh + 5h² + 4] - [5x² + 4]

------------------------------------------------- .

h

Let's simplify it a bit. Combine like terms up on top: The x²-terms disappear, as well as the 4's. (Remember to distribute the subtraction up on top; I almost forgot to do that while I was typing this!)

We end up with:

5h² + 10xh

----------------- .

h

Notice how there are h-terms on top? Both terms on top? That means we will be able to cancel out the h in the bottom. Factor the fraction like this:

5h(h + 2x)

------------------------- .

h

Cancel the h on top and bottom. We are left with 10x + 5h.