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

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SUMMARY

The equation of the tangent line to the curve y = x³ at the point P = (-3, -27) is determined by finding the slope (a) and y-intercept (b). The slope is calculated using the derivative, yielding a = 27. Subsequently, substituting the values into the tangent line equation y = ax + b allows for the calculation of b, resulting in b = 54. Therefore, the tangent line is expressed as y = 27x + 54.

PREREQUISITES
  • Understanding of calculus, specifically derivatives
  • Familiarity with the concept of tangent lines
  • Knowledge of polynomial functions, particularly cubic functions
  • Basic algebra skills for solving linear equations
NEXT STEPS
  • Study the rules of differentiation, focusing on polynomial derivatives
  • Learn how to find tangent lines for various functions
  • Explore applications of tangent lines in real-world scenarios
  • Practice solving problems involving cubic functions and their derivatives
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in understanding the principles of tangent lines and derivatives in polynomial functions.

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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