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

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Discussion Overview

The discussion revolves around finding the equation of a tangent line to the curve y = x³ at a specific point P = (-3, -27). Participants explore the process of determining the slope of the tangent line using derivatives and subsequently finding the y-intercept.

Discussion Character

  • Homework-related

Main Points Raised

  • One participant suggests that the slope (a) is the derivative of x³ evaluated at x = -3, proposing that a = 3(-3)².
  • Another participant agrees with the approach to find a and states that once a is known, it is straightforward to find b using the point P.
  • A later reply calculates a as 27 and attempts to find b using the equation y = 27(-3) + b, leading to a proposed value of b = 64.
  • Another participant corrects the arithmetic in the calculation of b, indicating that 81 - 27 equals 54, suggesting a potential error in the previous calculation.

Areas of Agreement / Disagreement

Participants generally agree on the method to find the slope and the subsequent steps to find b, but there is a disagreement regarding the correct value of b, as one participant points out an arithmetic error in the calculation.

Contextual Notes

The discussion includes unresolved mathematical steps, particularly in the calculation of b, which may depend on the interpretation of the arithmetic involved.

cgr4
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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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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 $b$.
 
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?
 
cgr4 said:
-27 = -81 + b

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

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