Mathematics - Introduction to Calculus Assignment -

Q1. An Earth-observing satellite can see only a portion of the Earth's surface. The satellite has horizon sensors that can direct the angle θ shown in the accompanying figure. Let r be the radius of the Earth (assumed spherical) and h the distance of the satellite from the Earth's surface.

a. Show that h = r(cos θ - 1).

b. Using = 6378 km, find the average rate of change of the distance from the satellite to the surface of the Earth, when θ changes from π /4 to π/3. What are the units?

c. At what rate is the distance h changing when θ = π/6?

Q2. Consider the graph of the function g shown below.

a. What is the domain of the function g?

b. Where is the function continuous?

c. Identify on the graph the local maximum.

d. Identify on the graph the local minimum.

e. Does it have an absolute maximum value? Explain.

f. Does it have an absolute minimum value? Explain.

Q3. Sketch the graph of the function f and its derivative function f' where

Q4. In each of the cases below, give the indicated derivative. You may not need to simplify your answers.

a. d/dx xcos√(x-3)

b. d²/x² x²tanx|_{x = π}

c. d/dx (sinx-cosx)/x³

d. If  f(o) =1, f′(o) =2,  g(0) = 0 and and g′(o) = -1, find

d/dx (f(x)=x²g(x))/(f(x)+g(x))|_x=0

Q5. A softball diamond is a square whose sides are 18 m long. Suppose that a player running from first to second base has a speed of 7.5 m/s at the instant she is 3 m from second base. At what rate is the player's distance from home plate changing at that instant?

Hint. Draw a diagram and locate the variables that change with respect to time.

Q6. Let f(x) = x-1/√x

a. Find the equation of the tangent line of the function f at the point (4, f(4)).

b. Use differentials to estimate the value of f(4.02).

Q7. Use implicit differentiation to prove that the curve x² + y² = (2x² + 2y² - x)² has a vertical tangent line at the point (1, 0).

Q8. One side of a right triangle is known to be 25 cm exactly. The angle opposite to this side is measured to be 60^o, with a possible error of ±0.5^o.

a. Use differentials to estimate the errors in the adjacent side and the hypothenuse.

b. Estimate the percentage errors in the adjacent side and hypothenuse.

Q9. Sketch the graph of one and only one function x $f$ which satisfies all the conditions listed below.

a. f(-x) = -f(x)

b. lim_x→4^- f(x) = ∞

c. lim_x→4^- f(x) = -∞

d. lim_x→∞ f(x) = 2

e. f′′(x) > 0 on the interval (0, 4)

Q10. Sketch the graph of the function

f(x) = (x²-4)/(x²+6)

Clearly indicate each of the steps.

Q11. The shoreline of a lake is a circle with diameter 3 km. Peter stands at point E and wants to reach the diametrically opposite point W. He intends to jog along the north shore to a point P and then swim the straight line distance to W. If he swims at a rate of 3 km/h and jogs at a rate of 24 km/h. How far should he jog in order to arrive at point W in the least amount of time?

12. a. Sketch the graphs of the curves y = sinx and y = x² showing their points of intersection.

b. Use the Intermediate Value Theorem to identify an interval where the equation sinx - x² = 0 has a non-zero solution.

c. Use Newton's method to approximate the non-zero solution of the equation sinx - x² = 0.