Calculus - angle between tangents

[rshen5]

Homework Statement

Find the angles between the tangents to y=-3e^-x and y=2+e^x at their poit of intersection

Homework Equations

y=-3e^-x and y=2+e^x

The Attempt at a Solution

i tried to find the point at which they intersect:
y=-3e^-x =2+e^x
where i got
x=ln 1
then i tried to find derivative of both equations:
y'=-3e^-x
y'=e^x
then subsitute the value of x in
and got
y'=-3
y'=1
forming 2 vectors ?
a=(-3,1)
b=(1,1)
then using vector = dot point equation to find the angle between them ..
a*b*cos (theta) = a_x*b_x+a_y+b_y

but i still got the wrong answer
the answe is : 63.43 degrees

 

[Mark44]

Homework Statement

Find the angles between the tangents to y=-3e^-x and y=2+e^x at their poit of intersection

Homework Equations

y=-3e^-x and y=2+e^x

The Attempt at a Solution

i tried to find the point at which they intersect:
y=-3e^-x =2+e^x
where i got
x=ln 1

ln 1 = 0

then i tried to find derivative of both equations:
y'=-3e^-x
y'=e^x

You should distinguish between the two different y values and y' values. For example, you could name them y₁ and y₂, and their derivatives y₁' and y₂'.

then subsitute the value of x in
and got
y'=-3
y'=1

Better, y₁' = -3 and y₂' = 1

forming 2 vectors ?
a=(-3,1)
b=(1,1)

Here is your mistake. Your a vector should have a slope of -3, but its slope is actually -1/3.

then using vector = dot point equation to find the angle between them ..
a*b*cos (theta) = a_x*b_x+a_y+b_y

but i still got the wrong answer
the answe is : 63.43 degrees

See above.

 

[rshen5]

Better, y₁' = -3 and y₂' = 1
Here is your mistake. Your a vector should have a slope of -3, but its slope is actually -1/3.
See above.

why is the slope (-1/3) not -3?

so does that mean in vector forms they will be:
a = (-1/3 , 1)
b= (1,1)

 

[Mark44]

why is the slope (-1/3) not -3?

Because the slope is rise/run. What is the slope of the line segment between (0, 0) and (-3, 1)?

so does that mean in vector forms they will be:
a = (-1/3 , 1)
b= (1,1)

Or a = (-1, 3)

 

[SteamKing]

rshen5:
This is a tricky problem since the two functions never intersect. See attached plot.

 

[Mark44]

rshen5:
This is a tricky problem since the two functions never intersect. See attached plot.

You're right. I didn't catch that minus sign in y = -3e^-x.