Problem 1: The velocity of an ant running along the edge of a shelf is modeled by the function
v(t) = 5t, 0 ≤ t < 1
6√, 1 ≤ t ≤ 2
where t is in seconds and v is in centimeters per second. Estimate the time at which the ant is 4 cm from its starting position.
Problem 2: Calculate the indefinite integrals listed below
a. ∫3x - 9/(√(x2-6x+1)) dx
b. ∫3 - tanθ/cos2θ.dθ
c. ∫(2 - x + x2)2/dx
d. ∫cos2 (3x) dx
Problem 3: Use the Mean Value Theorem to show that for any real numbers a, b
|cos a - cos b| ≤ a - b
Problem 4: Let f (x) = 3x3 + √x - 2 .
a. Find an interval where the function f has one root.
b. Use Rolle's Theorem to show that the function f has exactly one root.
Hint. See Example 2 on page 283 of the textbook.
Problem 5: Use the identity cos2 x + sin2 x = 1 to integrate ∫cos3x sin2x dx.
Problem 6: Evaluate each of the definite integrals listed below
a. 0∫Π/6 cos2(3x) dx
b. 0∫Π sin(2x)sin x dx
c. -2∫2 x2 + cos(2x) dx
Problem 7: Apply the fundamental theorem of calculus to find the following derivative
d/dx -x∫x2 tan(3t).dt
Problem 8: A circular swimming pool has a diameter of 24 ft., the sides are 5 ft. high, and the depth of the water is 4 ft.. How much work is required to pump all of the water out over the side? (Use the fact that water weighs 62.5 lb/ft3.)
Problem 9:
a. Sketch the region bounded by the curves y = 1/x2, y = 64x and y = 8x
b. Find the area of the region sketched in part a.
Problem 10: A motorcycle starting from rest, speeds up with a constant acceleration of 2.6 m/s2. After it has traveled 120 m, it slows down with a constant acceleration of -1.5 m/s until it attains a velocity of 12 m/s. What is the distance traveled by the motorcycle at that point?
Hint. Do not use decimals, leave square roots indicated.
Problem 11:
a. The temperature of a 10 m long metal bar is 15°C at one end and 30°C at the other end. Assuming that the temperature increases linearly from the cooler end to the hotter end, what is the average temperature of the bar?
b. Explain why there must be a point on the bar where the temperature is the same as the average, and find it.
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