Equation of the tangent line to the curve

In summary, to find an equation of the tangent line to the curve 7*x*e^(x)+8 at (0,8), take the derivative of the equation and calculate the slope by plugging in x=0. Then, use the slope-intercept form or point-slope form to find the final equation, which is y=7x+8.
  • #1
kdpointer
14
0

Homework Statement



Find an equation of the tangent line to the curve: 7*x*e^(x)+8 at (0,8)

Homework Equations



Derivative I guess?

The Attempt at a Solution



I know you have to take the derivative of the equation given which I think is

7*x*e^(x) + 7*e^(x)

Then you plug it into slope intercept form: y-y=m(x-x)

I did this and got y-8=(7*e^(x)*(x+1))*x, but apparently that's not right...

What did I do wrong? Maybe i just typed it in wrong?
 
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  • #2
Calculate the slope first by plugging in x=0 into the derivative equation... then when you have the number for the slope, plug it into the slope intercept form.

The slope should just be a number.
 
  • #3
Yeah, it's pretty wrong. In the line equation m is a constant. Otherwise, it wouldn't be a line, now would it? Put (0,8) into the derivative before you plug it into the line equation.
 
  • #4
so essentially you have to use your x-intercept twice
 
  • #5
So the slope is just 7?
 
  • #6
Thus the final equation is y-8 = 7x ?
 
  • #7
kdpointer said:
Thus the final equation is y-8 = 7x ?
it should probably be in slope-intercept form

y=mx+b
 
  • #8
It says that point-slope form is fine.. but it would be y=7x+8
 
  • #9
kdpointer said:
It says that point-slope form is fine.. but it would be y=7x+8

Yup. That's right.
 
  • #10
Alright.. thanks a lot!
 

FAQ: Equation of the tangent line to the curve

What is the equation of the tangent line to a curve at a given point?

The equation of the tangent line to a curve at a given point is given by the point-slope form: y - y1 = m(x - x1), where m is the slope of the tangent line and (x1, y1) is the given point on the curve.

How is the slope of the tangent line to a curve calculated?

The slope of the tangent line to a curve can be calculated using the derivative of the curve at the given point. The derivative represents the slope of the tangent line at any point on the curve.

Can the equation of the tangent line to a curve change at different points?

Yes, the equation of the tangent line to a curve can change at different points because the slope of the tangent line is dependent on the derivative, which can vary at different points on the curve.

Is the tangent line always perpendicular to the curve at the point of tangency?

No, the tangent line is not always perpendicular to the curve at the point of tangency. This only occurs when the curve is a circle or has a circular shape.

How is the equation of the tangent line to a curve used in real-world applications?

The equation of the tangent line to a curve is used in various real-world applications, such as physics, engineering, and economics. It helps in determining the instantaneous rate of change of a variable and can be used to predict future behavior of a system.

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