Question

Find the linear approximation L(x) of the function(x)=x^8+6x^2 at a=−1
L(x)= A + Bx where A=? and B=?

Answer 1

Alright, the first derivative is the slope. You can evaluate the function at a=-1 for (x,y) coordinates that you can use to find the linear approximation's y intercept.

Answer 2

how do i find A

Answer 3

Alright, step one. What's the derivative of f(x)? What does it mean?

Answer 4

8x^7 + 12x

Answer 5

What do you think f'(x) means?

Answer 6

derivative of f'(x)

Answer 7

...Yes, but it's the slope of f(x) at any given point x=a, right?

Answer 8

yes

Answer 9

So we already have a function y=f'(a)x+b; we just need to find b now. Do you follow?

Answer 10

yea. i found be to be -20. but my software isnt taking the derivative for A. it says it wants a number

Answer 11

Alright, A=f'(a). What did you get for that?

Answer 12

i have to take a second derivative?

Answer 13

No. I'll just run through this problem, then. So, we start with f(x)=x^8+6x^2, and we want to find the linear approximation at a=-1. Being that the approximation is going to be linear, we know the answer is going to be in the form y=mx+b, where m is the slope and b is the value of the approximation at x=0 (y-intercept).

The derivative of f(x), f'(x), gives me the slope of f(x) as a function of x. Since we want the slope at x=a, let's evaluate f'(a). f'(x) is 8x^7+12x, and thus, f'(a)=8a^7+12a. Since a is a constant, we have a constant value for slope at x=a, which means we have our m in y=mx+b→y=f'(a)x+b. So now all we're missing is b. Since the line's an approximation at only f(a), we need to evaluate f(a) such that we have a coordinate (a,f(a)). By plugging in the coordinates for our approximation into our slope...

f(a)=f'(a)a+b; solving for b here will give us our y-intercept.