Calculus - angle between tangents

In summary: Good catch!In summary, the problem is to find the angle between the tangents to the functions y=-3e^-x and y=2+e^x at their point of intersection. However, the two functions do not intersect and the correct answer is 63.43 degrees.
  • #1
rshen5
6
0

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
 
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  • #2
rshen5 said:

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
rshen5 said:
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'.
rshen5 said:
then subsitute the value of x in
and got
y'=-3
y'=1
Better, y1' = -3 and y2' = 1
rshen5 said:
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.
rshen5 said:
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.
 
  • #3
Mark44 said:
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)
 
  • #4
rshen5 said:
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)?
rshen5 said:
so does that mean in vector forms they will be:
a = (-1/3 , 1)
b= (1,1)
Or a = (-1, 3)
 
  • #5
rshen5:
This is a tricky problem since the two functions never intersect. See attached plot.
 

Attachments

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  • #6
SteamKing said:
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.
 

What is the definition of "angle between tangents" in calculus?

The angle between tangents in calculus refers to the angle formed between two tangent lines at a point on a curve. Tangent lines are lines that touch the curve at only one point, and the angle between them can be calculated using calculus principles.

How is the angle between tangents calculated in calculus?

The angle between tangents is calculated using the derivative of the curve at the point where the tangent lines intersect. This derivative represents the slope of the curve at that point, and the angle between tangents can be found using the formula tanθ = m1 - m2 / 1 + m1m2, where m1 and m2 are the slopes of the tangent lines.

Why is the angle between tangents important in calculus?

In calculus, the angle between tangents is important because it helps determine the behavior of a curve at a specific point. It can also be used to find the maximum and minimum values of a curve, as well as the points of inflection.

Can the angle between tangents be greater than 90 degrees?

Yes, the angle between tangents can be greater than 90 degrees. This can occur when the slopes of the tangent lines have opposite signs, resulting in a larger angle between them. It is also possible for the angle between tangents to be negative if the slopes have the same sign.

Are there any real-world applications of the angle between tangents in calculus?

Yes, the concept of the angle between tangents is used in various fields such as engineering, physics, and economics. It can be used to analyze the behavior of curves in real-world scenarios, such as predicting the trajectory of a projectile or understanding the supply and demand curves in economics.

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