Quotient Differences: Given f(x) = 5x^2 +2x, find [f(x+h)-f(x)] / h

Answer 1

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

f(x+h) = 5(x+h)^2 +2(x+h) = 5(x^2 +2xh +h^2) +2x +2h = 5x^2 +10xh +5h^2 +2x+2h

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

divide by h to get

10x +5h +2

a check on the answer is find the derivative of 5x^2 +2x = f'(x) = 10x +2

(f(x+h) - f(x))/h = f'(x) when h = 0

Answer 2

If you are having trouble plugging things into functions, first write it like:

[5(______)² + 2(______)]

then just put in whatever is asked of you ( "x+h" or "x" or whatever).

The top could look like

[5(______)² + 2(______)] - [5(______)² + 2(______)]

put x+h in the first, then x in the latter.

Answer 3

Hi! For this problem, you will need to go through several steps of substitution to evaluate your final answer as a quotient difference:

I always recommend taking note of our given information and "target" objective, so that we can quickly reference, when necessary. I find that it also helps me from feeling too overwhelmed by organizing the problem into a "problem and solution" flow that clearly shows my starting information that I can draw from, and how I should be utilizing it when I begin the substitution process.

Given: f(x) = 5x² + 2x

Find: f(x+h) - f(x)

----------------

h

Step 1: Substitute all f(x) x's with (x+h)

1. We're given that f(x) = 5x²+ 2x, therefore f(x+h) = [5(x+h)² +2(x+h)]

Step 2: Substitute 5x² + 2x for f(x)

1. f(x) = 5x² + 2x (for the remainder of the problem, we will refer to our given function in terms of x, as opposed to f(x). In this case, this means that we will use "5x²+ 2x" instead of "f(x)")

Step 3: Combine numerator functions

1. Numerator functions given in the problem are: f(x+h) - f(x)
2. In step 1, we found that f(x+h) = [5(x+h)²+2(x+h)]
3. In step 2, we found that f(x) = 5x²+ 2x
4. Therefore, f(x+h) - f(x) = [5(x+h)²+2(x+h)] - (5x²+ 2x)
5. *Remember to distribute the negative (-) when subtracting the full term.

Step 4: Distribute to simplify the term f(x+h), which we have written as [5(x+h)²+2(x+h)]

1. 5(x+h)²+2(x+h) = 5x² + 10xh + 5h²
2. 2(x+h) = 2x + 2h

Step 5: Write out the extended equation

1. Numerator: 5x² + 10xh + 5h² + 2x + 2h - 5x² - 2x
2. Since our goal is to find a fraction, we need to remember to include our denominator, "h," in the calculation
3. Denominator: h
4. Full fraction: (5x² + 10xh + 5h² + 2x + 2h - 5x² - 2x) / h

Step 6: Eliminate like-terms

1. Within the equation: (5x² + 10xh + 5h² + 2x + 2h - 5x² - 2x) / h we have 2 pairs of like terms:
2. 5x² - 5x² will cancel (remember, the sign that precedes the term must be considered. This is why we have -5x² as our second value, and not just 5x²)
3. 2x - 2x will also cancel-out.
4. By cancelling out the like-terms, we are left with: (10xh + 5h² + 2h) / h

Step 7: Factor common factor "h" in the numerator.

1. 10xh + 5h² + 2h --> h(10x+5h+2)

Step 8: Cancel out common factor "h" in numerator and "h" in the denominator

1. h(10x+5h+2) / h --> 10x+5h+2

Step 9: Record final solution to the problem

1. In our final step, we found that our equation simplified to 10x+5h+2. No further arithmetic can be performed, and so, we have reached our final answer.

Solution: 10x+5h+2