Please help me to explain these problems.
1) Suppose that total revenue is R(x) = 50x-0.5x^2 and total cost is C(x) = 10x+3 (both given in dollars), where x represents the number of units of a product that are produced and sold. Find the rate of change of the profit with respect to time when x = 10 units and dx/dt = 5 units per day.
(a) $200 per day
(b) $150 per day
(c) $80 per day
(d) $50 per day
2) A ladder 25 ft long leans against a vertical wall. If the lower end is being moved away from the wall at a rate of 6 ft/s, how fast is the height of the top changing (the answer will be a negative rate) when the lower end is 7 ft from the wall?
(a) -6 ft/s
(b) 1.75 ft/s
(c) -1.75 ft/s
(d) -2.083 ft/s
3) For f(x) = ex, which of the following statements is false?
(a) f′ is negative for all x.
(b) f′′(x) is positive for all x.
(c) The graph of f has no points of inflection.
(d) The graph of f is increasing.
4) 10. Which of the following statements is false?
(a) ln(AB) = ln(A) + ln(B)
(b) ln(A/B) = ln(A) - ln(B)
(c) e^ln(x) = x
(d) ln(x^r) = r ln(x^r-1)
5) For g(x) = ln(x), which of the following statements is false?
(a) The graph of g is concave down.
(b) g′′(x) is positive for all x in the domain of g.
(c) The graph of g has no relative extrema.
(d) The domain of g is (0,∞).
6) If y = e, then dy/dx is
(a) e
(b) 0
(c) 2.718
(d) None of the above.
7) If y = ln(3), then dy/dx is
(a) 1/3
(b) 1.10
(c) 0
(d) None of the above.
8) The concentration C, in parts per million, of a medication in the body t hours after ingestion is given by the function C(t) = 10t^2e^-t, where t ≥ 0. Find the maximum value of the concentration and the time at which it occurs.
(a) 5.4 ppm at t = 2 hr
(b) 4.61 ppm at t = 0.59 hr
(c) 2 ppm at t = 5.4 hr
(d) There is no maximum concentration.
Suppose that P0 = $1, 000 is invested in the Mandelbrot Bond Fund for which interest is compounded continuously at 5.9% per year. That is, the balance P grows at the rate given by dP/dt= 0.059P.
9) What is the balance after 2 years?
(a) $1,125.24
(b) $1,060.78
(c) $2,121.56
10) In about how many years will an investment of $1,000 double itself?
(a) 1.22 years
(b) 3.52 years
(c) 11.75 years