2.3 Homework: The Product and Quotient Rules

Transcription

1 2.3 Homework: The Product and Quotient Rules 1. Find the derivatives of each of the following functions. Label the derivative you find with its name using proper notation. a) f(x) = (x 2 x)3 x b) h(y) = y 3 sin y c) g(t) = t e t 1+2k d) m = 3 4k e) n = e θ (tan θ θ) f) p(y) = 1 sin y+cos y g) q(s) = s3 1 s 2 h) r(w) = 2 w (w + w w)

2 i) c(t) = t2 +2 t 4 3t 2 +1 j) a = 1 sin k 1 cos k k) b = 2z z sin z l) D(y) = y2 10y tan y m) u = b sin b b+1 n) v = xe x cos x

3 2. Find the equation for the tangent line to each of the curves given at the specified point. a) f(x) = 2x(2) x when x = 0 b) g(x) = 2x x 2 +1 when x = 1 3. Find m (t) if m(t) = t 4 sin t 4. If f(t) = tg(t), where g(3) = 4 and g (3) = 2, find an equation for the line tangent to f when t = 3.

4 5. The table below gives values of f(x) and g(x). Use it to help find the derivative of each function at the specified point. x f(x) g(x) f (x) g (x) / a) h(x) = f(x) + 2 x at x = 1 b) k(x) = x 2 g(x) at x = 2 c) m(x) = g(x) f(x)+1 at x = 1 d) n(x) = x g(x) at x = 2

5 6. Let f and g be differentiable functions for which the following information is known: f(2) = 5, g(2) = 3, f (2) = 1/2, g (2) = 2. a) Let h be the new function defined by the rule h(x) = f(x) g(x). Determine h(2) and h (2). b) Find an equation for the tangent line to y = h(x) at the point (2, h(2)). c) Let r be the function defined by the rule r(x) = f(x). Is r increasing, decreasing, or neither at a = 2? Why? g(x) d) Estimate the value of r(2.06) by using the tangent line at the point (2, r(2)).

6 7. f and g are given in the graphs below. Use the fact that h(x) = f(x)g(x), k(x) = f(x) g(x) and L(x) = g(x) along with the graphs to estimate: f(x) y=f (x ) 2 1 y=g (x ) a) h (1) b) k (1) c) h (2) d) k (2) e) L (1) f) L (2)

7 9. A farmer with large land holdings has historically grown a wide variety of crops. With the price of ethanol fuel rising, he decides that it would be prudent to devote more and more of his acreage to producing corn. As he grows more and more corn, he learns efficiencies that increase his yield per acre. In the present year, he used 7000 acres of his land to grow corn, and that land had an average yield of 170 bushels per acre. At the current time, he plans to increase his number of acres devoted to growing corn at a rate of 600 acres/year, and he expects that right now his average yield is increasing at a rate of 8 bushels per acre per year. Use this information to answer the following questions. a) Say that the present year is t = 0, that A(t) denotes the number of acres the farmer devotes to growing corn in year t, Y(t) represents the average yield in year t (measured in bushels per acre), and C(t) is the total number of bushels of corn the farmer produces. What is the formula for C(t) in terms of A(t) and Y(t)? b) What is the value of C(0)? What does it represent? c) Write an expression for C (t) in terms of A(t), A (t), Y(t), and Y (t). d) What is the value of C (0)? What does it represent? e) Based on the given information and your work above, estimate the value of C(1).

8 10. Let f(v) be the gas consumption (in liters/km) of a car going at velocity v (in km/hour). In other words, f(v) tells you how many liters of gas the car uses to go one kilometer if it is traveling at v kilometers per hour. In addition, suppose that f(80) = 0.05 and f (80) = a) Let g(v) be the distance the same car goes on one liter of gas at velocity v. What is the relationship between f(v) and g(v)? Use this to find g(80) and g (80) and explain their meaning. b) Let h(v) be the gas consumption in liters per hour of a car going at velocity v. In other words, h(v) tells you how many liters of gas the car uses in one hour if it is going at velocity v. What is the algebraic relationship between h(v) and f(v)? Use this to find h(80) and h (80) and explain their meaning. 11. The quantity sold, q, of a certain skateboard depends on the selling price p, in dollars, q = f(p). You are given that f(140) = 15,000 and f (140) = 100. a) What do f(140) = 15,000 and f (140) = 100 tell you about the sales of skateboards? b) The total revenue, R, earned by the sale of skateboards is given by R = pq. Find dr dp p=140 c) If the skateboards are currently selling for $140, what happens to revenue if the price is increased to $141?

9 12. A museum has decided to sell one of its paintings and to invest the proceeds. If the picture is sold between the years 2015 and 2035 and the money from the sale is invested in a bank account earning 5% interest per year compounded annually, then B(t), the balance in the year 2035 depends on the year, t, in which the painting is sold and the sale price P(t). If t is measured from the year 2015 so that 0 t 20 then a) Explain why this formula for B(t) makes sense. B(t) = P(t)(1.05) 20 t b) Show how the formula for B(t) can be rewritten as: B(t) = (1.05) 20 P(t) (1.05) t c) If P(10) = 150,000 and P (10) = 5000, find B (10) and explain its meaning. 13. In 2013, the population in the Miami-Ft Lauderdale metropolitan area was 3,354,000 and was increasing at roughly 45,000 people per year. The average annual income in the area was $31,107 and was increasing by about $1900 per year. Estimate the rate at which the total personal income was rising, in the area in 2013.