Graph the function f(x) = x^2 + 3x and draw the tangent...

Graph the function f(x) = x^2 + 3x and draw the tangent lines to the graph at points whose x-coordinates are -2, 0, and 1. Find the difference quotient (f(x+h) - f(x))/h Find f'(x) by determining lim h->0 (f(x+h) - f(x))/h. Find f'(-2) (This slope should match the tangent line you drew above.) Find f'(0) Find f'(1)

Transcript

00:01 Well, to start this problem, i'm going to do the difference quotients where they have, let me just write out what f of x is, this x squared plus 3x, and then just get into the difference quotients where what you have, i mean, they even tell you what to do.

00:19 You have the h on bottom, and then this f of x goes right here, x squared plus 3x.

00:25 And on this f of x plus h, you're taking out the x's.

00:31 And you're replacing those xes with x plus h.

00:35 Now, i would not write that as my final answer, which might be why that was marked wrong.

00:40 I would go ahead and foil out and distribute.

00:44 You have x squared plus 2xh plus 3h plus 3h minus x squared.

00:51 Oh, i've forgotten h squared in here.

00:53 Sorry about that.

00:55 Minus 3x all over h.

00:58 And what you have there is, let's see, x squared's cancel, the 3x cancels.

01:04 And i wouldn't even write this as my next answer because in the numerator you can cancel out the h's, or factor out the h's, and you can cancel this out, this h out, and this would be my answer for that first part.

01:22 So then in part b, what you have to do next, is substitute 0 in for this h.

01:30 And 0, 2x plus 0 plus 3 is just 2x plus 3.

01:35 So that would be my answer for that.

01:37 I'll just call this a, this b.

01:41 So now we can actually get into the, like, evaluating f prime of negative 2, because i can substitute negative 2 into this.

01:49 So that's negative 4 plus 3 is negative 1.

01:52 And then f prime of 0.

01:54 Well, plug in 0 and for x, and you get 0.

01:58 0 plus 3 is 3.

02:00 And then you also have to do f prime of 1, and you end up with 2 times 1 is 2 plus 3 is 5...

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