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AP Calculus AB: 3.2.1 The Slope of a Tangent Line

Mathematics9 CardsCreated 3 months ago

This flashcard set details how to find the slope of a tangent line using the derivative. It covers the step-by-step process from the definition of the derivative to evaluating the derivative at a specific point, with multiple examples illustrating how to calculate slopes of tangent lines for various functions.

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The Slope of a Tangent Line

  • To find the slope of a tangent line, evaluate the derivative at the point of tangency.

  • The derivative of f at x is given by provided the limit exists.

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Key Terms

Term
Definition

The Slope of a Tangent Line

  • To find the slope of a tangent line, evaluate the derivative at the point of tangency.

  • The derivative of f at x is given by ...

note

  • To find the slope of a line tangent to a curve at a given point, it is necessary to take the derivative.

  • Start with the defi...

Using the definition of the derivative, find the slope of the tangent line to the function f (x) = 12x ^2 at (−2, 48).

−48

Consider the function f(x)=−1/3x^3−x.Suppose you are given that f′(x)=−x^2−1.What is the slope of the tangent line to f(x) at (3,−12)?

−10

Given f(x)=2x^2−x, what is the slope of the line tangent to f(x) at the point (3,15)?

m = 11

Given f(x)=x^2−2x and f′(x)=2x−2,what is the slope of the line tangent to f(x) at the point (1,−1)?

m = 0

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AP Calculus AB: 3.2.1 The Slope of a Tangent Line
TermDefinition

The Slope of a Tangent Line

  • To find the slope of a tangent line, evaluate the derivative at the point of tangency.

  • The derivative of f at x is given by provided the limit exists.

note

  • To find the slope of a line tangent to a curve at a given point, it is necessary to take the derivative.

  • Start with the definition of the derivative.

  • Substitute the function into the definition.

  • Expand the expression so you can find pieces that cancel.

  • Every term that does not have a xshould cancel away.

  • Factor a x out of the remaining expression.

  • Cancel the xwith the one in the denominator.

  • Now evaluate the resulting limit by direct substitution.

  • The resulting equation is the derivative of the function f. Notice that the derivative is not the answer to the question. There is more work to do.

  • Now that you know the derivative of f, find the slope of the tangent line by plugging the point of tangency into the derivative.

  • The resulting number is the slope of the tangent line.

  • The derivative gives you the slope.

Using the definition of the derivative, find the slope of the tangent line to the function f (x) = 12x ^2 at (−2, 48).

−48

Consider the function f(x)=−1/3x^3−x.Suppose you are given that f′(x)=−x^2−1.What is the slope of the tangent line to f(x) at (3,−12)?

−10

Given f(x)=2x^2−x, what is the slope of the line tangent to f(x) at the point (3,15)?

m = 11

Given f(x)=x^2−2x and f′(x)=2x−2,what is the slope of the line tangent to f(x) at the point (1,−1)?

m = 0

Given f  (x) = x^2 − 2, what is the slope of the line tangent to f (x) at the point (3, 7)?

m = 6

Suppose you are given f(x)=x^3 and f′(x)=3x^2.Find the value of x,1

√21/3

Given f(x)=x^2+x, what is the slope of the line tangent to f(x) at the point (2,  6)?

m = 5