AP Calculus AB: 3.2.3 The Equation of a Tangent Line
To find the tangent line’s equation, first find the derivative (slope function), then evaluate it at the point of tangency. Use the slope and point in the point-slope form to write the line. Horizontal tangents occur where the derivative equals zero.
The Equation of a Tangent Line
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.
Key Terms
The Equation of a Tangent Line
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 sub...
note
To find the equation of a line tangent to a curve, start by taking the derivative.
Remember, the derivative evaluated at a p...
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).
g ′ (−1) = 1/2
At what point is the slope of the line tangent to the curve y=3x^2+4 equal to zero?
(0, 4)
Suppose f(x)=x^2−3. What is the equation of the line tangent to the curve with a slope equal to 2?
y+2=2(x−1)
At what point is the slope of the line tangent to the curve y=x^2−2x+1 equal to 2?
(2, 1)
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| Term | Definition |
|---|---|
The Equation of a Tangent Line | 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. |
note |
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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). | g ′ (−1) = 1/2 |
At what point is the slope of the line tangent to the curve y=3x^2+4 equal to zero? | (0, 4) |
Suppose f(x)=x^2−3. What is the equation of the line tangent to the curve with a slope equal to 2? | y+2=2(x−1) |
At what point is the slope of the line tangent to the curve y=x^2−2x+1 equal to 2? | (2, 1) |
Suppose f (x) = −x^ 2 + 4. What is the equation of the line tangent to the curve at the point (−1, 3)? | y = 2x + 5 |
What is the equation of the line tangent to the curve f(x)=3x^2−2at the point (−2,10)? | y−10=−12(x+2) |
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. | (1/2,−1/4) |
What is the equation of the line tangent to the curve y = x^ 2 − 2x + 1 at (3, 4)? | y − 4 = 4 (x − 3) |
Find the equation of the line tangent to the curve y = 3x ^2 + 4 when x = 3. | y − 31 = 18 (x − 3) |
Suppose f(x)=x^2−3x.What is the equation of the line tangent to the curve with a slope equal to −1? | y=−x−1 |
What is the equation of the horizontal tangent line to the curve f(x)=3x^2−2? |
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Suppose f(x)=−x^2+2.What is the equation of the line tangent to the curve with a slope equal to 1? | y=x+9/4 |
Suppose f(x)=x^2−3x.What is the equation of the line tangent to the curve at the point (2,−2)? | y=x−4 |
Suppose f (x) = x ^3 . What is the equation of the line tangent to the curve at the point (−1, −1)? | y = 3x + 2 |
Find the equation of the line tangent to the graph of f (x) = x ^2 − 5 at (2, −1). | y = 4x − 9 |