Question

Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. (If an answer does not exist, enter DNE.)
y = −6x + e^x

Answer 1

Tangent line? How is that related to "derivative?"

Answer 2

Does any of your learning material discuss how the derivative of a function represents the slope of the tangent line to the graph?

Answer 3

Find the derivative with respect to x of y = −6x + e^x .

The result, dy/dx, represents the slope of the tangent line to y = −6x + e^x.

Please get started by finding this dy/dx.

Answer 4

dy/dx=-6+e^x

Answer 5

Yes. Now, at what point on the graph of y = −6x + e^x do you want to find the slope of the tangent line?

Answer 6

Hint: Go back and re-read the original question.

Answer 7

it doesn't give me any points
But i have to find the zeros of this so i set the dy/dx = 0

Answer 8

so -6+e^x = 0

Answer 9

x = ln(6)?

Answer 10

Pls explain further: why do you have to set the deriv. equal to zero?

Answer 11

so that I can find the x coordinate and then I can plug that back into the original eqution

Answer 12

"Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line." First we want to know whether the graph actually has a horiz. tan. line (slope = 0).
It does, because ... ????

Answer 13

the correct anwser is (x,y) = ( ln(6), 6-6ln(6) )
which I have no idea how they got to that except for the ln (6)

Answer 14

I was hoping you'd answer my question.
How do we know that the graph of this function has a horiz. tangent line?
In other words, do we know that the slope of the T. L. to this graph is zero anywhere?

Answer 15

no I do not know if there is a zero anywhere

Answer 16

Hint: The slope of the t. l. is given by dy/dx = -6 + e^x.

Answer 17

In other words, do we know that the slope of the T. L. to this graph is zero anywhere?

Answer 18

Answer: Yes. Question for you: Why?

Answer 19

I do not know

Answer 20

minimum?

Answer 21

Because dy/dx = -6 + e^x =0 has a ..... ? (no, it's not minimum). Try just once more, please.

Answer 22

From before: "In other words, do we know that the slope of the T. L. to this graph is zero anywhere?"

Answer 23

The slope is dy/dx = -6 + e^x, and we have to set this = to 0 because of our search for a possible horiz. tan. line.

Answer 24

Does this equation have a solution?

Answer 25

yes
ln(6)

Answer 26

?

Answer 27

Can you solve -6+e^x=0 for x?

Yes. There is a solution. And therefore, the graph has a horiz. tan line.

Now, please write out the original function.

Answer 28

y=-6x+e^x

Answer 29

Good. We found that there's a horiz. t. l. when x = ln 6.

Please find f(ln 6)...evaluate the orignl fun. at x = ln 6.

Answer 30

-6(ln6)+e^ln(6)

Answer 31

Write the coordinates of this point in the form (x, y), please. Also recall that e^(ln 6) = 6.

Answer 32

e^x and ln x are inverse functions.

Answer 33

y=6-6*ln6
so ( ln (6) , 6-6ln(6) )
( x , y )

Answer 34

There is a horiz. tan line to the graph of f(x)=-6x+e^x at the point you have given.
Does this match the answer?

Answer 35

Remember: the correct answer, once we've determined that there is a horiz. tan. line, are the coordinates of the point of tangency.

Answer 36

yes it is

Answer 37

*yes it does match my answer

Answer 38

Cool. thanks for your perseverance. All done.

Answer 39

Thank you for your help

Answer 40

I wanted to emphasize that "the derivative at any point on a curve is the slope of the tangent line to the curve at that point."

You're welcome!