How Do You Find the Equation of a Tangent Line at a Given Point?

So $b=54$.In summary, the line y = ax + b is tangent to y=x3 at the point P = (-3,-27). To find a, we use the derivative of x3 which is 3x^2. Substituting x=-3, we get a=27. We can then use this value to find b by plugging in the coordinates of the point P into the equation y = ax + b, giving us b=54.
  • #1
cgr4
7
0
The line y = ax + b is tangent to y=x3 at the point P = (-3,-27). Find a and b.

I'm pretty lost on this one.

Here's my initial thoughts. Find (a) first.

(a) is equal to the slope which is the derivative of x3

So (a) would be equal to 3(-3)2 ?

Not quite sure where to start on this one.
 
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  • #2
cgr4 said:
The line y = ax + b is tangent to y=x3 at the point P = (-3,-27). Find a and b.

(a) is equal to the slope which is the derivative of x3

So (a) would be equal to 3(-3)2 ?
Yes. Once you know $a$, $y=-27$ and $x=-3$, it is easy to find $b$.
 
  • #3
Evgeny.Makarov said:
Yes. Once you know $a$, $y=-27$ and $x=-3$, it is easy to find $b$.

so a = 27

so, y = 27(-3) + b

-27 = -81 + b

therefore b = 64?
 
  • #4
cgr4 said:
-27 = -81 + b

therefore b = 64?
Correct, except that 81 - 27 = 54.
 

FAQ: How Do You Find the Equation of a Tangent Line at a Given Point?

What is the equation of a tangent line?

The equation of a tangent line is a linear equation that represents the slope of a curve at a specific point. It is written in the form 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 need to know the coordinates of the point where the line touches the curve. Then, you can use the formula y - y1 = m(x - x1), where m is the slope of the tangent line and (x1, y1) are the coordinates of the point.

What does the slope of a tangent line represent?

The slope of a tangent line represents the rate of change of the curve at a specific point. It shows how much the y-value changes for every unit change in the x-value.

Can the equation of a tangent line change?

Yes, the equation of a tangent line can change depending on the point where it is being drawn on the curve. Each point on a curve has a different tangent line with a unique equation.

How is the equation of a tangent line used in calculus?

The equation of a tangent line is used in calculus to find the derivative of a function at a specific point. The derivative represents the slope of the tangent line at that point and can be used to determine the rate of change of the function.

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