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Problem 4.2.34
An inverse tangent identity
b. Prove that tan⁻¹ x + tan⁻¹ x(1/x) = π/2, for x > 0.
Problem 4.2.38
Find all functions whose derivative is f'(x) = x + 1.
Problem 4.2.45
Running pace Explain why if a runner completes a 6.2-mi (10-km) race in 32 min, then he must have been running at exactly 11 mi/hr at least twice in the race. Assume the runner’s speed at the finish line is zero.
Problem 10
Let ƒ(x) = x²⸍³ . Show that there is no value of c in the interval (-1, 8) for which ƒ' (c) = (ƒ(8) - ƒ (-1)) / (8 - (-1)) and explain why this does not violate the Mean Value Theorem.
Problem 11
11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.
ƒ(x) = x (x - 1)² ; [0, 1]
Problem 12
11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.
ƒ(x) = sin 2x; [0, π/2]
Problem 13a
Let ƒ(x) = (x - 3) (x + 3)²
a. Verify that ƒ'(x) = 3(x - 1) (x + 3) and ƒ"(x) = 6 (x + 1).
Problem 13d
Let ƒ(x) = (x - 3) (x + 3)²
d. Determine the intervals on which ƒ is concave up or concave down.
Problem 13e
Let ƒ(x) = (x - 3) (x + 3)²
e. Identify the local extreme values and inflection points of ƒ .
Problem 13f
Let ƒ(x) = (x - 3) (x + 3)²
f. State the x- and y-intercepts of the graph of ƒ.
Problem 14
11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.
ƒ(x) = 1 - | x | ; [-1, 1]
Problem 15
11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.
ƒ(x) = 1 - x²⸍³ ; [-1, 1]
Problem 17
11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.
g(x) = x³ - x² - 5x - 3; [-1, 3]
Problem 19
Lapse rates in the atmosphere Refer to Example 2. Concurrent measurements indicate that at an elevation of 6.1 km, the temperature is -10.3° C and at an elevation of 3.2km , the temperature is 8.0°C . Based on the Mean Value Theorem, can you conclude that the lapse rate exceeds the threshold value of 7°C/ km at some intermediate elevation? Explain.
Problem 20
Drag racer acceleration The fastest drag racers can reach a speed of 330 mi/hr over a quarter-mile strip in 4.45 seconds (from a standing start). Complete the following sentence about such a drag racer: At some point during the race, the maximum acceleration of the drag racer is at least _____ mi/hr/s. .
Problem 21
21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].
b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.
ƒ(x) = 7 -x² ; [-1; 2]
Problem 23
21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].
b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.
ƒ(x) = { - 2x if x < 0 ; x if x ≥ 0 ; [-1, 1]
Problem 26
21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].
b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.
ƒ(x) = ln 2x; [1,e]
Problem 30
21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].
b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.
ƒ(x) = x + 1/x; [1,3]
Problem 47
Mean Value Theorem for quadratic functions Consider the quadratic function f(x) = Ax² + Bx + C, where A, B, and C are real numbers with A ≠ 0. Show that when the Mean Value Theorem is applied to f on the interval [a,b], the number guaranteed by the theorem is the midpoint of the interval.
Problem 48
Means
b. Show that the point guaranteed to exist by the Mean Value Theorem for f(x) = 1/x on [a,b], where 0 < a < b, is the geometric mean of a and b; that is, c = √ab.
Problem 49
Equal derivatives Verify that the functions f(x) = tan² x and g(x) = sec² x have the same derivative. What can you say about the difference f - g? Explain.
Problem 50
100-m speed The Jamaican sprinter Usain Bolt set a world record of 9.58 s in the 100-meter dash in the summer of 2009. Did his speed ever exceed 30 km/hr during the race? Explain.
Problem 51
Verify the identity sec⁻¹ x = cos⁻¹ (1/x), for x ≠ 0.
Problem 53
Suppose f'(x) < 2, for all x ≥ 2, and f(2) = 7. Show that f(4) < 11.
Problem 64
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
f(x) = -x⁴ - 2x³ + 12x²