3.2.1 The Slope of a Tangent Line

Question 1

The Slope of a Tangent Line

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

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

Answer 3

−48

Question 4

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)?

Answer 4

−10

Question 5

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

Answer 5

m = 11

Question 6

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)?

Answer 6

m = 0

Question 7

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

Answer 7

m = 6

Question 8

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

Answer 8

√21/3

Question 9

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

Answer 9

m = 5