Math 180, Calculus I Final Exam December 4, 1995

(25 pts) 1. Find the derivatives of the following functions. Do not simplify your answer. (i) ln(1 + x^{2} + x^{4})

(ii) sin(x^{2}) + 3 cos x
(iii) \frac{e^{−3x}}{1 + x^{2}}
(iv) arctan(3x)
(v) axe^{bx}

(20 pts) 2. If P dollars are invested at an annual rate of r%, then after t years this investment

grows to F dollars, where

F = P (1 + \frac{r}{100})^{t}.

(i) Find \frac{dF}{dr} . (Assume P and t are constant.)

(ii) What does the statement \frac{dF}{dr} |_{r=5} = 160 tell you about the cost or benefit of an increase in the annual rate r?

(15 pts) 3. Find \frac{dy}{dx} in terms of x and y when x and y are related by the equation

y^{3} + xy + 6 = 0.

(15 pts) 4. Evaluate the integral: ∫_{0}^{a} cos x dx.

(15 pts) 5. Find the area above the x-axis, under the curve y = \sqrt{x}, and between x = 0 and x = 9.

(20 pts) 6. A car travelling on a gravel road had a flat tire and stopped unevenly. The table below shows its velocity (in feet per second) t seconds after the flat.

t in sec 0 1 2 3 4 5 6 7 8
v in ft/sec 66 60 52 40 28 14 6 0 0

(i) Approximately how far did the car go in the first second? (ii) Approximately how much time did it take the car to stop? (iii) Approximately how far had the car gone when it came to a stop?

(25 pts) 7. Let f(x) = x^{3} − x^{2} − x + 1.

(i) Find and classify all the critical points of f.

(ii) Find all the inflection points.

(iii) Sketch the graph of f on the interval −1.5 ≤ x ≤ 1.5.

(iv) Find the global maximum and minimum of f(x) for −1.5 ≤ x ≤ 1.5.

(25 pts) 8. The graph below represents the rate of change of a function f with respect to x, that is, it is the graph of f′.

The graph of f′(x)

(i) On what intervals is f increasing or decreasing?

(ii) On what intervals is the graph of f concave up? Concave down?

(iii) You are given that f(0) = 0. Is there another number a with 0 < a < 2 for which f(a) = 0? If not, explain why not, and if so, give an approximate value of a.

(20 pts) 9. Let f(x) = tan x − 0.5.

(i) Find the equation of the tangent line to y = f(x) at x = π/4.

(ii) Where does this tangent line intersect the x-axis?

(20 pts) 10. A landscape architect plans to enclose a 3000 square-foot rectangle in a botanical garden. She will use shrubs costing $25 per foot along three sides and fencing costing $20 per foot along the fourth side. Find the dimensions that minimize the total cost.