MATH 142 Business Mathematics II

Transcription

1 MATH 142 Business Mathematics II Summer, 2016, WEEK 2 JoungDong Kim Week 2: 4.1, 4.2, 4.3, 4.4, 4.5 Chapter 4 Rules for the Derivative Section 4.1 Derivatives of Powers, Exponents, and Sums Differentiation Formulas 1. Constant rule: If f(x) = c where c is any constant then f (x) = 0 2. Power Rule: If f(x) = x n, where n is a real number, then f (x) = nx n 1 3. Constant times a function rule: (cf(x)) = cf (x) 4. Sum/Difference rule: If f(x) = g(x)±h(x), then f (x) = g (x)±h (x) 1

2 Ex1) Differentiate the following functions: a) f(x) = 5 b) f(x) = π c) f(x) = e 2 d) f(x) = x 7 e) f(x) = x 1 2 f) f(x) = 3 x 5 g) f(x) = 5 x 2 h) y = 2x 3 +5x 9 i) g(t) = 3t t 1 2 +e j) h(x) = x3 +2x 2 2x+3 x k) y = x4 +4x 2 3 x 2

3 Ex2) If JD drop the ball from a building 400 feet tall, its height above the ground (in feet) after t seconds is given by s(t) = t 2 a) Compute s (t) b) Compute s(2) and s (2) Ex3) If f(x) = 3x 4 2x 2, where does the graph of the function have a horizontal tangent line? 3

4 Ex4) Suppose the total cost (in dollars) of producing x books is given by a) Find C(15) C(14) C(x) = 0.5x 2 12x+100 b) Find C (14) Marginal Business Functions Approximate change in the dependent variable (cost, revenue, profit) when the independent variable (the number of items produced/sold) is changed by a single unit. Marginal Cost Function Marginal Revenue Function Marginal Profit Function MC(x) = C (x) MR(x) = R (x) MP(x) = P (x) NOTE. The marginal functions approximate the Cost/Revenue/Profit of the next item. 4

5 Ex5) The total profit (in dollars) of producing x ski jackets is given by P(x) = 0.2x x a) Find the exact profit realized from the sale of the 201st ski jacket. b) Use the marginal profit function to approximate the profit realized from the sale of the 201st ski jacket. 5

6 Derivatives of Exponential and Logarithmic Functions 1. If f(x) = e x then f (x) = e x 2. If f(x) = lnx then f (x) = 1 x 3. If f(x) = b x then 4. If f(x) = log b x then f (x) = b x lnb f (x) = 1 xlnb Ex6) Find the derivative of each of the following functions: a) f(x) = 7e x b) f(x) = 2(3) x c) f(x) = lnx+3 d) f(x) = 4x 2 3 x+log 7 x 3 5 x e) y = ln(x 7 )+3(2) x Ex7) Find the equation of the line tangent to the graph of f(x) = e x +lnx at x = 1. 6

7 Section 4.2 Derivatives of Products and Quotients Product Rule If h(x) = f(x) g(x) and if f (x) and g (x) exist, then h (x) = f (x) g(x)+f(x) g (x) Quotient Rule If h(x) = f(x) g(x) and if f (x) and g (x) exist, then h (x) = f (x) g(x) f(x) g (x) [g(x)] 2 Ex8) Find the derivative of the following functions: a) h(x) = x 2 (x 2 +4x) b) h(x) = (x 2 +3)( 4 x+ 8 x 3 ) 7

8 c) h(x) = x2 +5 3x d) h(x) = 3 x+7x x 2 4x+ 1 x e) f(x) = 5x 4 e x 8

9 f) g(x) = x2 e x +5 7 e x g) h(x) = 2x +5x 2 log 2 x 4 lnx 9

10 Ex9) Find the equation of the tangent line to f(x) = x x 3 4x 2 +2 at x = 2. 10

11 Ex10) Suppose that f(2) = 1, g(2) = 3, f (2) = 4, and g (2) = 6. Find h (2) for each of the following: a) h(x) = 2f(x) 3g(x) b) h(x) = f(x)g(x) c) h(x) = f(x) g(x) d) h(x) = f(x) 1+g(x) 11

12 Ex11) Let P(x) = F(x)G(x) and Q(x) = F(x), where F and G are the functions whose graphs are G(x) shown below. y 5 4 F(x) G(x) x a) Find P (2) b) Find Q (7) 12

13 Section 4.3, 4.4 The Chain Rule The Chain Rule: If g is differentiable at x and f is differentiable at g(x), then the composite function F(x) = f(g(x)) is differentiable at x and is given by F (x) = f (g(x)) g (x) In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then dy dx = dy du du dx General Derivative Rules If y = [f(x)] n then If y = e f(x) then If y = ln(f(x)) then If y = b f(x) then If y = log b f(x) then y = n[f(x)] n 1 f (x) y = e f(x) f (x) y = 1 f(x) f (x) y = b f(x) lnb f (x) y = 1 f(x) lnb f (x) 13

14 Ex12) Differentiate the following: a) f(x) = (4x 2 +7x) 5 b) g(x) = 6(x 1 2 3x) 4 c) y = 3 (t 2 +3t+4) 4 d) F(x) = e x2 e) H(x) = 3x 5 e x4 f) h(x) = 3 4 (3x+2) 5 14

15 g) f(x) = (4x 2 +5) 6 ( 3 (3x 4 5x+7) 4 ) h) y = log 8 ( 1+x 2 +10x) ( ) x 1 i) y = log 6 x+2 j) y = 5ln((x2 +x) 5 ) x 3 15

16 Ex13) Find the value(s) of x where the tangent line is horizontal for f(x) = x 2 (2 3x) 3 16

17 Ex14) Suppose w(x) = u(v(x))andu(0) = 1, v(0) = 2, u (0) = 3, u (2) = 4, v (0) = 5, andv (2) = 6. Find w (0). Ex15) Let y = lnu and u = 5x 4 +x 6. Find dy dx. Ex16) Keith invests $5,000 into a savings account offering interest at an annual rate of 2.4% compounded continuously. How fast is the balance growing after 8 years? 17

18 Section 4.5 Elasticity of Demand Elasticity of demand is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price. Price elasticities are almost always negative, although analysts tend to ignore the sign. Explore and Discuss) A broker is trying to sell you two stocks: Biotech and Comstat. The broker estimates that Biotech s price per share will increase $2 per year over the next several years, while Comstat s price per share will increase only $1 per year. Is this sufficient information for you to choose between the two stokes? What other information might you request from the broker to help you decide? If Biotech costs $100 a share and Comstat costs $25 a share, then which stock is the better buy? To answer this question, we introduce two new concepts: relative rate of change and percentage rate of change. Definition. Relative and Percentage Rates of Change The relative rate of change of a function f(x) is f (x) f(x), or equivalently, d dx lnf(x). The Percentage rate of change is 100 f (x) d, or equivalently, 100 f(x) dx lnf(x). 18

19 Ex17) What are their relative and percentage rates of change? Definition. Elasticity of Demand: Let the price p and demand x for a product be related by a price-demand equation of the form x = f(p). Then the elasticity of demand at price p, denoted by E(p), is relative rate of change of demand E(p) = relative rate of change of price Theorem. Elasticity of Demand: Ifpriceanddemandarerelatedbyx = f(p), thentheelasticity of demand is given by E(p) = pf (p) f(p) 19

20 Ex18) The price p and the demand x for a product are related by the price-demand equation x+500p = 10,000. Find the elasticity of demand, E(p), and interpret each of the following: (a) E(4) (b) E(16) (c) E(10) In Sum, E(p) Demand Interpretation Revenue 0 < E(p) < 1 Inelastic Demand is not sensitive to changes in price, that is, percentage change in price produces a smaller percentage change in demand. E(p) > 1 Elastic Demand is sensitive to changes in price, that is, a percentage change in price produces a larger percentage change in demand. E(p) = 1 Unit A percentage change in price produces the same percentage change in demand. A price increase will increase revenue. A price increse will decrease revenue. Revenue is maximized here. 20

21 Ex19) If E(p) = 2.5 and price is decreased by 2%, what happens to demand? Does the revenue increase or decrease? Ex20) Given that x = f(p) = 1875 p 2, determine whether demand is elastic, inelastic, or has unit elasticity at the indicated values of p. a) p = 15 b) p = 25 c) p = 40 21

22 Ex21) The price-demand equation for home-delivered 12-inch pizzas is p = x where x is the number of pizzas delivered weekly. The current price of one pizza is $8. In order to generate additional revenue from the sale of 12-inch pizzas, would you recommend a price increase or a price decrease? 22