For the function ƒ(x) = x^3 - 8x + 3, calculate where a = 5 and h = 2. Simplify before substitution (the simplifying is where I am having problems...not sure if my answer is right). Thanks.
Using the limits equation.
Knowing what a and h are is what is confusing me. They are not defined and do not appear in f(x)
yes....f(a+h)-fa)/h
OK, [f(a+h)-f(a)]/h is the approximation to the derivative at x = 5. It is the average slope of the function from f(5) to f(7).
Perhaps they want it expressed in terms of a and h. In that case,
[f(a+h)-f(a)]/h =
[a+h)^3 -a^3 -8(a+h)+a]/h
= [a^3 + 3a^2h + 3ah^2 + h^3 -a^3 -8h]/h
= 3a^2 + 3 ah + h^2 -8
in the limit as h goes to 0, this becomes the derivative 3a^2 -8, or the value of f'(x) at x = a.
To calculate ƒ(a + h) for the function ƒ(x) = x^3 - 8x + 3, where a = 5 and h = 2, we need to simplify the expression before substituting the values.
The expression ƒ(a + h) means we need to replace every x in the function with (a + h).
So, let's simplify the function by substituting (a + h) for x:
ƒ(a + h) = (a + h)^3 - 8(a + h) + 3
Now, expand and simplify the expression. To do this, we need to use the binomial expansion formula:
(a + h)^3 = a^3 + 3a^2h + 3ah^2 + h^3
Substituting this expansion into our expression:
ƒ(a + h) = (a^3 + 3a^2h + 3ah^2 + h^3) - 8(a + h) + 3
Next, distribute the -8 across (a + h):
ƒ(a + h) = a^3 + 3a^2h + 3ah^2 + h^3 - 8a - 8h + 3
Now, we can substitute the values of a = 5 and h = 2 into this expression:
ƒ(5 + 2) = 5^3 + 3(5^2)(2) + 3(5)(2^2) + 2^3 - 8(5) - 8(2) + 3
Now, compute each term:
ƒ(5 + 2) = 125 + 3(25)(2) + 3(5)(4) + 8 - 40 - 16 + 3
ƒ(5 + 2) = 125 + 150 + 60 + 8 - 40 - 16 + 3
Finally, add and subtract these numbers to get the final result:
ƒ(5 + 2) = 290
So, ƒ(a + h) = 290 when a = 5 and h = 2.
Remember, the key steps to follow are:
1. Replace every x in the function with (a + h).
2. Expand and simplify the expression.
3. Distribute any constants or coefficients across the terms.
4. Substitute the values of a and h.
5. Compute each term.
6. Add and subtract to get the final result.