Mathematics - Introduction to Calculus Assignment -
Q1. The velocity of an ant running along the edge of a shelf is modeled by the function
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.
Q2. Calculate the indefinite integrals listed below
a. ∫(3x-9)/√(x2-6x+1) dx
b. ∫(3-tanθ)/cos2θ dθ
c. ∫(2-x+x2)2/√x dx
d. ∫cos2(3x) dx
Q3. Use the Mean Value Theorem to show that for any real numbers a, b
|cos a - cos b| ≤ |a - b|
Q4. 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.
Q5. Use the identity cos2x + sin2x = 1 to integrate ∫cos3xsin2x dx.
6. Evaluate each of the definite integrals listed below
a. 0∫π cos2(3x) dx
b. 0∫π sin(2x) sinx dx
c. -2∫2 x2 + cos(2x) dx
Q7. Apply the fundamental theorem of calculus to find the following derivative d/dx -x∫x^2 tan(3t) dt.
Q8. 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.)
Q9. 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.
Q10. 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.
Q11. a. The temperature of a 10 m long metal bar is 15oC at one end and 30oC 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.