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Finding tangent line equations using the formal definition of a limit

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.B (LO)
,
CHA‑2.B.2 (EK)
,
CHA‑2.B.3 (EK)
,
CHA‑2.B.4 (EK)
,
CHA‑2.C (LO)
,
CHA‑2.C.1 (EK)
This structured practice takes you through three examples of finding the equation of the line tangent to a curve at a specific point.
We can calculate the​ slope of a tangent line using the definition of the derivative of a function f at x, equals, c (provided that limit exists):
limit, start subscript, h, \to, 0, end subscript, start fraction, f, left parenthesis, c, plus, h, right parenthesis, minus, f, left parenthesis, c, right parenthesis, divided by, h, end fraction
​Once we've got the slope, we can ​find the equation of the line. This article walks through three examples.
Function f is graphed. The positive x-axis includes value c. The graph is a curve. The curve starts in quadrant 2, moves downward to a point in quadrant 1, moves upward through a point at x = c, and ends in quadrant 1. A tangent line starts in quadrant 4, moves upward, touches the curve at the point at x = c, and ends in quadrant 1.

Example 1: Finding the equation of the line tangent to the graph of f, left parenthesis, x, right parenthesis, equals, x, squared at x, equals, 3

Step 1
What's an expression for the derivative of f, left parenthesis, x, right parenthesis, equals, x, squared at x, equals, 3?
Choose 1 answer:
Choose 1 answer:

Step 2
Evaluate the correct limit from the previous step.
f, prime, left parenthesis, 3, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

f, prime, left parenthesis, 3, right parenthesis gives us the slope of the tangent line. To find the complete equation, we need a point the line goes through.
Usually, that point will be the point where the tangent line touches the graph of f.
Step 3
What is the point we should use for the equation of the line?
left parenthesis
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis

Step 4
Complete the equation of the line tangent to the graph of f, left parenthesis, x, right parenthesis, equals, x, squared at x, equals, 3.
y, equals

And we're done! Using the definition of the derivative, we were able to find the equation for the line tangent to the graph of f, left parenthesis, x, right parenthesis, equals, x, squared at x, equals, 3.
Function f is graphed. The x-axis goes from negative 12 to 12. The graph is a U-shaped curve. The curve starts in quadrant 2, moves downward to (0, 0), moves upward through a point at about (3, 9), and ends in quadrant 1. A tangent line starts in quadrant 4, moves upward, touches the curve at the point, and ends in quadrant 1.

Example 2: Finding the equation of the line tangent to the graph of g, left parenthesis, x, right parenthesis, equals, x, cubed at x, equals, minus, 1

Step 1
g, prime, left parenthesis, minus, 1, right parenthesis, equals, question mark
Choose 1 answer:
Choose 1 answer:

Example 3: Finding the equation of the line tangent to f, left parenthesis, x, right parenthesis, equals, x, squared, plus, 3 at x, equals, minus, 5

Let's do this one without all the steps.
What is the equation of the tangent line?

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