Expert: Sherman D. - 3/20/2004

Question
Find and simplify:

{f(a+h)-f(a)}/h

1. f(x)=4x-3
2. f(x)=4x^2-7x+6
3. f(x)=x^3
4. f(x)=square root of x

Thanks...I am almost getting these types of questions. It seems that I am not understanding the order of operations in algebra or more likely, the factoring out of the brackets.  Anyway, I look forward to your answers, with explanations maybe as well? Bye.

Answer
Please = Parenthesis, Brackets
Excuse = Exponents
My = Multiply
Dear = Divide
Aunt = Add
Sally = Subtract

{f(a + h) - f(a)}/h


1.)
f(x) = 4x - 3

f(a + h) = 4(a + h) - 3
f(a + h) = 4a + 4h - 3

f(a) = 4a - 3

Substitute the values for your functions, and you get

{(4a + 4h - 3) - (4a - 3)}/h
{4a + 4h - 3 - 4a + 3}/h
{4h}/h

Answer : 4

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{f(a + h) - f(a)}/h

2.
f(x) = 4x^2 - 7x + 6

f(a + h) = 4(a + h)^2 - 7(a + h) + 6
f(a + h) = 4((a + h)(a + h)) - 7a - 7h + 6
f(a + h) = 4(a^2 + ah + ah + h^2) - 7a - 7h + 6
f(a + h) = 4(a^2 + 2ah + h^2) - 7a - 7h + 6
f(a + h) = 4a^2 + 8ah + 4h^2 - 7a - 7h + 6

f(a) = 4a^2 - 7a + 6

Substitute and you get

{f(a + h) - f(a)}/h

{(4a^2 + 8ah + 4h^2 - 7a - 7h + 6) - (4a^2 - 7a + 6)}/h
{4a^2 + 8ah + 4h^2 - 7a - 7h + 6 - 4a^2 + 7a - 6}/h
{8ah + 4h^2 - 7h}/h

Answer : 8a + 4h - 7

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{f(a + h) - f(a)}/h

3.)
f(x) = x^3

f(a + h) = (a + h)^3
f(a + h) = (a + h)(a + h)(a + h)
f(a + h) = (a^2 + ah + ah + h^2)(a + h)
f(a + h) = (a^2 + 2ah + h^2)(a + h)
f(a + h) = (a^3 + a^2h + 2a^2h + 2ah^2 + ah^2 + h^3)
f(a + h) = a^3 + 3a^2h + 3ah^2 + h^3

f(a) = a^3

Substitute and you get

{(a^3 + 3a^2h + 3ah^2 + h^3) - (a^3)}/h
{a^3 + 3a^2h + 3ah^2 + h^3 - a^3}/h
{3a^2h + 3ah^2 + h^3}/h

Answer : 3a^2 + 3ah + h^2

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{f(a + h) - f(a)}/h

4.)
f(x) = sqrt(x)

f(a + h) = sqrt(a + h)

f(a) = sqrt(a)

substitute and you get

{(sqrt(a + h)) - sqrt(a)}/h

That is as far as i got, even if you factor it, you can't take anything out.

As for explaination, all i can say is to substitute "a + h" for "x" for the individial problems given, then substitute "a" for the problem given. Then substitute those values that you get for "a + h" and "a" into your main problem.