I don't understand how to do the folliwing problems.
Find and simplify the difference quotient f(x+h) -f(x)/h, h cannot equal 0 for the given formula.
1. f(x)=5x^2
options: a. 5, b.5(2x+h), c. 10/h +x+5h, d. 5(2x^2+2xh+h^2)/h
2. f(x)=x^2+3x+9
options: a. 2x+h+9, b.1, c.2x^2+2x+2xh+h^2+h+18/h, d. 2x+h+3
Here is how to do #1. #2 is the same.
f(x+h) = 5 (x+h)^2 = 5x^2 + 10 xh + 5h^2
f(x+h) - f(x) = 10xh + 5h^2
[f(x+h) - f(x)]/h = 10 x + 5h = 5x (2+h)
This is answewr # (b)
As h -> 0, the ratio approaches 10 x
To find and simplify the difference quotient, we need to substitute the given function into the formula f(x+h) - f(x) / h and simplify the expression.
1. Let's start with the first problem:
f(x) = 5x^2
Substitute f(x+h) into the formula:
f(x+h) = 5(x+h)^2
Expand (x+h)^2:
f(x+h) = 5(x^2 + 2xh + h^2)
Now, substitute the values into the formula:
f(x+h) - f(x) = 5(x^2 + 2xh + h^2) - 5x^2
Simplify the expression by distributing 5:
f(x+h) - f(x) = 5x^2 + 10xh + 5h^2 - 5x^2
Combining like terms, we get:
f(x+h) - f(x) = 10xh + 5h^2
Finally, divide the expression by h:
(f(x+h) - f(x)) / h = (10xh + 5h^2) / h
Simplifying further, we get:
(f(x+h) - f(x)) / h = 10x + 5h
Therefore, the correct option for the first problem is b. 5(2x + h).
2. Now, let's move on to the second problem:
f(x) = x^2 + 3x + 9
Substitute f(x+h):
f(x+h) = (x+h)^2 + 3(x+h) + 9
Expand and simplify (x+h)^2 and distribute:
f(x+h) = x^2 + 2xh + h^2 + 3x + 3h + 9
Substitute into the formula:
f(x+h) - f(x) = (x^2 + 2xh + h^2 + 3x + 3h + 9) - (x^2 + 3x + 9)
Simplify the expression by canceling out like terms:
f(x+h) - f(x) = x^2 + 2xh + h^2 + 3x + 3h + 9 - x^2 - 3x - 9
Combining like terms, we get:
f(x+h) - f(x) = 2xh + h^2 + 3h
Divide the expression by h:
(f(x+h) - f(x)) / h = (2xh + h^2 + 3h) / h
Simplifying further:
(f(x+h) - f(x)) / h = 2x + h + 3
Therefore, the correct option for the second problem is c. 2x^2 + 2x + 2xh + h^2 + h + 18/h.
Remember, to find and simplify the difference quotient, you substitute the given function into the formula and then simplify the expression step by step.