Flashcards in 3.2.1 The Slope of a Tangent Line Deck (9):
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.
- 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).
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)?
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