3.2.3 The Equation of a Tangent Line

Question 1

The Equation of a Tangent Line

Answer 1

• To find the equation of a line tangent to a curve, take the derivative, evaluate the derivative at the point of tangency to find the slope, and substitute the slope and the point of tangency into the point-slope form of a line. • To find where the line tangent to a curve is horizontal, set the derivative equal to zero and solve for x.

Question 2

note

Answer 2

• To find the equation of a line tangent to a curve, start by taking the derivative.
• Remember, the derivative evaluated at a point gives you the slope of the line tangent to the curve at that point.
• Plug the point of tangency into the derivative. The result is the slope of the tangent line.
• The equation of a line requires two pieces of information: the slope and a point on the line. Find the y-coordinate of the point of tangency by substituting the x-value into the function f.
• Remember, the point of tangency is on the tangent line because it is the point where the line touches the curve.
• Substitute the slope and thex- and y-coordinates into the point-slope form of a line to get the equation of the tangent line.
• Horizontal tangents lead to many applications of calculus.
• The derivative is a function machine that produces slopes. If you want to know where the slope of the line tangent to the curve equals zero, you must set the derivative equal to zero and find the x-value that makes that statement true.

Question 3

Suppose you are told that the equation of the line tangent to the graph of a function g(x)at (−1,−2) is y=1/2x−3/2. Find g′(−1).

Answer 3

g ′ (−1) = 1/2

Question 4

At what point is the slope of the line tangent to the curve y=3x^2+4 equal to zero?

Answer 4

(0, 4)

Question 5

Suppose f(x)=x^2−3. What is the equation of the line tangent to the curve with a slope equal to 2?

Answer 5

y+2=2(x−1)

Question 6

At what point is the slope of the line tangent to the curve y=x^2−2x+1 equal to 2?

Answer 6

(2, 1)

Question 7

Suppose f (x) = −x^ 2 + 4. What is the equation of the line tangent to the curve at the point (−1, 3)?

Answer 7

y = 2x + 5

Question 8

What is the equation of the line tangent to the curve f(x)=3x^2−2at the point (−2,10)?

Answer 8

y−10=−12(x+2)

Question 9

Consider the function f (x) = x^ 2 − x. Using the fact that f ′ (x) = 2x − 1, find the point (x, y) on the graph of f (x) where the tangent line is a horizontal line.

Answer 9

(1/2,−1/4)

Question 10

What is the equation of the line tangent to the curve y = x^ 2 − 2x + 1 at (3, 4)?

Answer 10

y − 4 = 4 (x − 3)

Question 11

Find the equation of the line tangent to the curve y = 3x ^2 + 4 when x = 3.

Answer 11

y − 31 = 18 (x − 3)

Question 12

Suppose f(x)=x^2−3x.What is the equation of the line tangent to the curve with a slope equal to −1?

Answer 12

y=−x−1

Question 13

What is the equation of the horizontal tangent line to the curve f(x)=3x^2−2?

Answer 13

y = −2

Question 14

Suppose f(x)=−x^2+2.What is the equation of the line tangent to the curve with a slope equal to 1?

Answer 14

y=x+9/4

Question 15

Suppose f(x)=x^2−3x.What is the equation of the line tangent to the curve at the point (2,−2)?

Answer 15

y=x−4

Question 16

Suppose f (x) = x ^3 . What is the equation of the line tangent to the curve at the point (−1, −1)?

Answer 16

y = 3x + 2

Question 17

Find the equation of the line tangent to the graph of f (x) = x ^2 − 5 at (2, −1).

Answer 17

y = 4x − 9