Question

Given f(x)=(3x-1)^2 Find the equation of the line tangent to the graph x=2, and find the value of x where the tangent line is horizontal

Answer 1

http://bit.ly/QU4mW2

Answer 2

f(x) = (3x-1)^2

f ' (x) = 2(3x-1)^1 ... use the power rule

now you need to multiply that with the derivative of the inside, so you need to derive 3x-1 to get 3, then you multiply 3 by that last expression on the right side to get

f ' (x) = 3*2(3x - 1)^1

which simplifies to

f ' (x) = 6(3x - 1)

f ' (x) = 18x - 6

Answer 3

now plug in x = 2 into f ' (x) and tell me what you get

Answer 4

30

Answer 5

that's the slope of the tangent line at x = 2

Answer 6

so m = 2

Answer 7

I meant, m = 30

Answer 8

now we need to find the y coordinate of the point on f(x) when x = 2

f(x) = (3x-1)^2

f(2) = (3(2)-1)^2 ... plug in x = 2

f(2) = (6-1)^2

f(2) = 5^2

f(2) =25

Answer 9

so the point (2,25) is on the function f(x)

Answer 10

we have enough to find the equation of the tangent line at x = 2

this is because we have a slope (m = 30) and a point the line goes through (2,25)

Answer 11

y = mx + b

y = 30x + b

25 = 30(2) + b

solve for b

Answer 12

tell me what you get

Answer 13

-35

Answer 14

so your tangent line is

y = 30x - 35

Answer 15

tangent line at x = 2

Answer 16

that answers the first part of the question

Answer 17

f ' (x) = 18x - 6

represents the slope of the tangent line at ANY point on f(x)

Answer 18

if the tangent line is horizontal, then the slope is 0

Answer 19

so when we want to find what values of x make horizontal tangent lines, we just plug in f ' (x) = 0 and solve for x




f ' (x) = 18x - 6

0 = 18x - 6

...

...

x = ??

Answer 20

3

Answer 21

close

Answer 22

.3

Answer 23

but it's a bit off

Answer 24

repeating

Answer 25

good or x = 1/3

Answer 26

at x = 1/3, the tangent line is horizontal

Answer 27

finding horizontal tangents is useful for maximizing or minimizing a function

so you can apply it to maximizing profit or minimizing costs

Answer 28

well...that seems very easy

Answer 29

yeah it's not too bad once you get to know it

Answer 30

that seemed like algebra easy...like I was WAY overthinking it!

Answer 31

well you're definitely using calculus ideas and rules, but in a way, yes it's like algebra