Question

For the function f(x)=3x^2 at the point (4,48), find the a) slope of the curve, b) an equation of the tangent line, c) an equation of the normal line.

Answer 1

To find the slope of the curve at a point, we can use calculus to help us.

Answer 2

The derivative of f(x) is f '(x) = 6x. To find the slope at the point (4, 48), plug the x-coordinate into f '(x).

Answer 3

f '(4) = 6 * (4) = ...?

Answer 4

Are you there? @tambam16

Answer 5

yup listening

Answer 6

Alright, I would like you to try participating. What is 5 * 4 =...?

Answer 7

Sorry 6 * 4 = ...?

Answer 8

so it would be f(x)=6(4) is 24.. and?

Answer 9

24 is the slope of the of the curve at (4, 48)

Answer 10

This means the answer to part a is 24.

Answer 11

oh k

Answer 12

Alright, are you ready for part b?

Answer 13

yes yes

Answer 14

To find the equation of a tangent line we can use the slope we got from part a (24), and the point (4, 48) by plugging them into the "point-slope form" of a line: y - y1 = m(x - x1).

Answer 15

m = slope, x1 = 4, y1 = 48

Answer 16

y - 48 = 24(x - 4)

Answer 17

Can you try simplifying that equation?

Answer 18

y=24x-48

Answer 19

That's correct! The answer to part b is y = 24x - 48.

Answer 20

Ready for part c?

Answer 21

yeah

Answer 22

Okay, before we begin part c, I would like to check your understanding of what "normal" means in math. Are you familiar with the meaning of "normal"?

Answer 23

umm...not really

Answer 24

Okay, normal means perpendicular. Are you familiar with that word?

Answer 25

oh yea. so the slope would be -1/24

Answer 26

Very good! That is correct. Now use the new slope in the same way we did for part b. Plug the slope and the point into the equation y - y1 = m(x - x1)

Answer 27

m = -1/24, x1 = 4, y1 = 48

Answer 28

Can you please try plugging that into the equation?

Answer 29

ok..this is a weird number but it's y=-1/24x + 289/6

Answer 30

y - 48 = (-1/24)(x - 4)

Answer 31

y = (-1/24)x + 1/6 + 288/6

Answer 32

y = (-1/24)x + 289/6

Answer 33

You got it correct!

Answer 34

thanks @cebroski !! you are amazing!

Answer 35

You are welcome! Have a great rest of your night!