Find the equation of the tangent line pt 3 (Need someone to check my work)

In summary, the equation of a tangent line is y = mx + b, where m is the slope and b is the y-intercept. To find the equation of a tangent line, you take the derivative of the function at the given point, then use the point-slope form of a line to simplify to y = mx + b. The point-slope form of a line is y - y<sub>1</sub> = m(x - x<sub>1</sub>). A tangent line can only intersect a curve at one point, and a curve can have a vertical tangent line when the slope at a certain point is undefined.
  • #1
shamieh
539
0
find the equation of the tangent line.\(\displaystyle y = e^{3x + cos x}\) @ x = 0

\(\displaystyle
y1 = e^{3(0) + cos(0)} = e^1 = e\)
\(\displaystyle y1 = e\)

\(\displaystyle y = e^{3x+cos x}\)
\(\displaystyle y' = e^{3x + cos x} * (3 - sinx)\)

\(\displaystyle m = 3e\)

\(\displaystyle
y - e = 3e(x - 0) \)
 
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  • #2
shamieh said:
find the equation of the tangent line.\(\displaystyle y = e^{3x + cos x}\) @ x = 0

\(\displaystyle
y1 = e^{3(0) + cos(0)} = e^1 = e\)
\(\displaystyle y1 = e\)

\(\displaystyle y = e^{3x+cos x}\)
\(\displaystyle y' = e^{3x + cos x} * (3 - sinx)\)

\(\displaystyle m = 3e\)

\(\displaystyle
y - e = 3e(x - 0) \)
Looks good to me!

-Dan
 
  • #3
Thank God! Was so worried about the \(\displaystyle e^1 = e \)
 

FAQ: Find the equation of the tangent line pt 3 (Need someone to check my work)

What is the equation of a tangent line?

The equation of a tangent line is y = mx + b, where m is the slope of the tangent line and b is the y-intercept.

How do you find the equation of a tangent line?

To find the equation of a tangent line, you first need to find the slope of the tangent line at the given point. This can be done by taking the derivative of the function at that point. Then, plug in the slope and the given point into the point-slope form of a line, y - y1 = m(x - x1). Simplify the equation to get it in the standard form, y = mx + b.

What is the point-slope form of a line?

The point-slope form of a line is y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line.

Can a tangent line intersect a curve at more than one point?

No, a tangent line can only intersect a curve at one point. This is because a tangent line is defined as a line that touches the curve at one point and has the same slope as the curve at that point. If it intersected the curve at more than one point, it would have a different slope at each point, which would not be a tangent line.

Is it possible for a curve to have a vertical tangent line?

Yes, it is possible for a curve to have a vertical tangent line. This occurs when the slope of the curve at a certain point is undefined, which happens when the derivative of the curve at that point is equal to 0. In this case, the equation of the tangent line would be x = a, where a is the x-coordinate of the point of tangency.

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