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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.1 • Rates of Change and the Derivative 175

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AP EXAM TIP • Read the AP Exam Tip on the left. Following the tip, we would use the interval
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of least width containing the number c. So, on the AP Exam, we would use the
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To earn points on the AP Exam, when interval [1, 2] to approximate the derivative f (c). Based on the result in the
′
approximating the derivative of a function f ′
second bullet, we have f (2) ≈ 9.
at a number c using a table, the interval of
least width containing c must be used. In Approximating the Derivative of a Function Represented
other words, use the interval that most tightly EXAMPLE 9
by a Table
bounds c.
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The table below lists several values of a function y = f (x) that is continuous on the
closed interval [0, 6] and has a derivative at each number in the open interval (0, 6).
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AP EXAM TIP
Approximate the derivative of f at 3.4.
Problems similar to Example 9 often appear
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on the AP Exam. x 1 2 3 4 5
f (x) 4 6 2 1 6
Solution
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AP EXAM TIP
Note that 3.4 is not in the table, so we do not know f (3.4). In such cases, approximate
If you are asked to find an approximation of
the derivative by finding the average rate of change using the interval of least width
the derivative f (c) of a function y = f (x)
′
containing 3.4. In this case, find the average rate of change from 3 to 4, namely,
represented by a table and c is not in the
table, use the interval of least width
f (4) − f (3) 1 − 2
′
containing c to approximate f (c). = = −1
4 − 3 1
Then f (3.4) is approximately −1.
′
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NOW WORK Problem 51 and AP Practice Problems 8 and 9.
2.1 Assess Your Understanding
Concepts and Vocabulary
1 √
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1. True or False The derivative is used to find instantaneous 169 11. f (x) = at (1, 1) 12. f (x) = x at (4, 2)
velocity. x

2. True or False The derivative can be used to find the rate of 1 at 1, 1 2 at 1, 2
13. f (x) = 14. f (x) =
change of a function. x + 5 6 x + 4 5
3. The notation f (c) is read f of c; f (c) represents PAGE 1 1
′
′
174 15. f (x) = √ at (1, 1) 16. f (x) = 2 at (1, 1)
the of the tangent line to the graph of f at the point . x x
f (x) − f (3)
4. True or False If it exists, lim is the derivative of In Problems 17–20, find the rate of change of f at the indicated
x→3 x − 3
numbers.
the function f at 3.
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170 17. f (x) = 5x − 2 at (a) c = 0, (b) c = 2
′
5. If f (x) = 6x − 3, then f (3) = .
2
18. f (x) = x − 1 at (a) c = −1, (b) c = 1
6. The velocity of an object, the slope of a tangent line, and the rate 2
of change of a function are three different interpretations of the 19. f (x) = x at (a) c = 0, (b) c = 1
mathematical concept called the . x + 3
x
20. f (x) = 2 at (a) c = 0, (b) c = 2
Skill Building x − 1
In Problems 21–30, find the derivative of each function at the given
In Problems 7–16,
number.
(a) Find an equation for the tangent line to the graph of each 21. f (x) = 4x + 1 at 1 22. f (x) = 5x − 9 at 2
function at the indicated point.
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(b) Find an equation of the normal line to each function at the 173 23. f (x) = x − 2 at 0 24. f (x) = 2x + 4 at 1
indicated point. 25. f (x) = 3x + x + 5 at −1 26. f (x) = 2x − x − 7 at −1
2
2
(c) Graph the function, the tangent line, and the normal line at
√ 1
the indicated point on the same set of coordinate axes. 27. f (x) = x at 4 28. f (x) = at 2
x 2
2
2
7. f (x) = 3x at (−2, 12) 8. f (x) = x + 2 at (−1, 3)
2 − 5x 2 + 3x
3
3
9. f (x) = x at (−2, −8) 10. f (x) = x + 1 at (1, 2) 29. f (x) = 1 + x at 0 30. f (x) = 2 + x at 1
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