Math 110 Final Exam General Review. Edward Yu

Transcription

1 Math 110 Final Exam General Review Edward Yu

2 Da Game Plan Solving Limits Regular limits Indeterminate Form Approach Infinities One sided limits/discontinuity Derivatives Power Rule Product/Quotient Rule Chain Rule

3 Marginal Functions MR, MC, MP, Elasticity Implicit Differentiation & Related Rates Application of the 1 st Derivative Intervals of Increasing and Decreasing Relative Extremas

4 Application of the 2 nd Derivative Concavity Inflection Points Law of Diminishing Returns The second derivative test Optimization a.k.a Absolute Max/Min Word Problems!!!

5 Exponential Functions Logarithmic Functions Ln, e. Compound Interests General Compound Interest Continuous Compound Interest Effective Interest Rate Present Value General compounded (periodic) Continuous compounded (e)

6 Evaluating Limits Polynomials -> just substitute the limit in. Ex: lim x 2 3x2 + x + 5 =3( 2) = = 15

7 Special case Ex: lim x 2 x 2 4 x 2 = 0 0 Indeterminate form

8 Indeterminate Form Factoring lim x 2 x 2 4 x 2 Rationalizing (Conjugate) lim x 4 x 4 x 2 L'Hopital's Rule

9 Comparison Factoring lim x 2 x 2 4 x 2 =lim x 2 (x+2)(x 2) x 2 =lim x 2 x + 2 = 4 Rationalization lim x 4 x 4 x 2 =lim x 4 x 4 x 2 =lim x+2 x+2 (x 4)( x+2) x 4 x 4 =lim x 4 x + 2 = 4 conjugate a b a + b = a 2 + b 2

10 Comparison Factoring lim x 2 x 2 4 x 2 =lim x 2 (x+2)(x 2) x 2 =lim x 2 x + 2 = 4 L'Hopital's Rule lim x 2 x 2 4 x 2 = lim x 2 (x 2 4) (x 2) =lim x 2 2x 1 =2(2) = 4

11 Limits approaching Infinites For Polynomials Ex: lim x 5x 3 1 ALWAYS DNE!!! For Rational Functions lim x ± f(x) g(x) Three Cases Degree of TOP is Bigger Degree of BOTTOM is Bigger Same

12 Degree of Top > Degree of Bottom Case 1: The highest power is in the numerator. Always DNE (± ). Ex: lim 2x 5 x 3x lim x x 3 2 x 4 = DNE or = DNE or

13 Degree of Bottom > Degree of Top Case 2: The highest power is in the denominator Always = 0 Ex: lim x 4x 1 3x 4 +2 = 0

14 Degree of Bottom = Degree of Top Case 3: The highest power is same on the top and the bottom. limit is the quotient of the coefficients. Ex: lim x 2x 3 +7x 4 5x 3 2x = 2 5

15 Solving for Discontinuities Goal: Turn a discontinued function into differentiable everywhere. Key: Sub in the restriction of x! 3x 4 if x 2 Ex: let x 2, what value of k + kx if x > 2 will make the function continuous everywhere? 3x 4 = x 2 + kx, while x = = k = k

16 Derivatives 1. Power Rule x n 2. Product Rule f(x)g(x) 3. Quotient Rule f x g x 4. Chain Rule f x n

17 Marginal Analysis -It s the change analysis via the 1 st derivative!! C(x) -> Total Cost R(x) -> Total Revenue P(x) -> Total Profit C (x) -> Marginal Cost R (x) -> Marginal Revenue P (x) -> Marginal Profit Cost = Revenue Profit C(x) = R(x) P(x) R(x) = xf(x) f(x) is the price or demand function

18 Marginal Revenue Question Suppose the demand for computer speakers is given by P x = 300 x, where x is the number of speakers, and p x is the unit cost in dollars. What s the marginal revenue if 50 units were sold? Marginal Revenue <- Revenue <- P(x)

19 Solution R(x) = p(x) x R x = 300 x x R x = 300x x 2 R x = 300 2x Since 50 units were sold.. R 50 = 300 2(50) R 50 = $200 the approximate revenue of selling the 51 st unit is $200

20 Elasticity Elasticity of demand: E p = Pf (p) f(p) Watch OUT!! In a typical demand function, P = f x Switch into, x = f P

21 Cause & Effect Causes Demand is Elastic Demand is Inelastic Effects If E(p)>1 -> small % change in price causes greater % change in demand If E(p)<1 -> small% change in price causes a even smaller % change in demand Demand is unitary If E(p)=1 -> same amount!!

22 Example Staples has determined that the demand for erasers is given by: P = 6 x. P is the unit price in 20 dollars and x is the number of erasers sold per day. What s the price elasticity when the price is at $2 per piece? What does the number mean? Hint: rearrange and solve for x

23 P = 6 x 20 x 20 = 6 p x = p = f(p) Remember E p f x = 20 E 2 = = 1 2 Inelastic. = Pf (p) f(p)

24 Implicit Differentiation y 3 + 7y = x 3 Two-Step Problem to find dy or dx y. 1) Differentiate both sides in respect of x. When differentiating y, put a y or dy after y dx 2) Solve the resulting function for y or dy in terms of x and y. dx

25 Let s solve it! y 3 + 7y = x 3 3y 2 y + 7(1)y = 3x 2 y 3y = 3x 2 y = 3x2 3y 2 +7

26 1 st and 2 nd Derivative Application Increasing & Decreasing Intervals Local Extremas 1 st Derivative Slope Tangent

27 1 st and 2 nd Derivative Application Concavity Inflection Points 2 nd Derivative Point of Diminishing Return

28 Approach: Increasing/Decreasing Intervals 1. Take derivative (finding slopes of the function) f (x). 2. Set f 0 x = undefined 3. Create open intervals by these points. 4. Select a test point C in each interval and determine the sign of f (c) 5. Intervals + increasing decreasing

29 Exercise f x = 2 3 x3 2x 2 6x 2, where is f(x) increasing/decreasing? f x = 2x 2 4x 6 0 = 2x 2 4x 6 0 = 2 x 2 2x 3 0 = x 3 x + 1 x = 3 1

30 Continued f x = (x 3)(x + 1) f 2 = = 5 (positive) f 0 = = 3 (negative) f 5 = = 12 (positive) Increase on, 1 (3, ) Decrease on ( 1,3)

31 Process for finding Relative Extremas 1. Find critical points. (Domain!) 2. Determine the sign of f (x) to the left and right on each critical point. + Local Max + Local Min + + NOTHING! NOTHING!

32 Concavities f x > 0, the curve is concave up f x < 0, the curve is concave down

33 Law of Diminishing Return -point beyond which there are smaller & smaller returns for each $ invested. Application of the second derivative From concave up to concave down

34 Visually In Economics In Math 110

35 Optimization Goal: Find absolute Max/Min over an interval a, b 1. Find the critical points in the interval a, b 2. Compute the value of f at each critical point and check endpoints f(a), f(b) 3. Largest Absolute Max Smallest Absolute Min

36 f x = x x 2 +1 max/min Quick Exercise, over 0,2 Find absolute f x = x2 +1 2x(x) x f x = 1 x2 x = 0 x = 1; x = 1 X=-1 not in the interval 0,2

37 1 x = 0 2 f x = x x 2 +1 f 1 = 1 2 f 0 = 0 Absolute Max Absolute Min f 2 = 2 5

38 Log/Ln Functions Since Log/Exponential Functions are inverses: 2 most important rules! e lnx = x, x > 0 lne x = x

39 Tricky Sample Exam Questions Solve for x: A) ln lnx = 1 e ln lnx = e 1 lnx = e e lnx = e e x = e e

40 Solve for x B) e ex = 10 lne ex = ln10 e x = ln10 lne x = ln ln10 x = ln ln10

41 Interests Problems General Compound Interest mt A = P 1 + r m Continuous Compound Interest A = Pe rt Effective Interest Rate (EFF) r eff = 1 + r 1 m Present Values General compounded (periodic) Continuous compounded (e) m

42 Effective Interest Rates Ex. Find the effective rate of interest corresponding to: a) 10%/year compounded semi-annually b) 9%/year compounded quarterly a) r eff = b) r eff = = 10.25%/year 4 1 = 9.308%/year

43 Summing up A = P A = Pe rt 1 + r m mt r eff = 1 + r m m 1 P = A 1 + r m P = Ae rt mt (Present Value) (Present Value)

44 TIME TO GET REAL

45 2008 Fall Exam 13. Find the derivative of f t = e 2t ln (t + 1). a) 2e2t t+1 b) e2t t+1 + 2e2t ln t + 1 c) 2e 2t + 1 t+1 d) e2t t+1 + e2t ln (t + 1)

46 2008 Fall Exam 10. Simplify ln 2e 5x + ln e3x 2 a)8x b)e 8x c)ln2 ln x d) e 5x 2 + e 3x 1 2

47 2008 Fall Exam 14. Find the derivative of f x = ln x 2 4 a) b) c) d) 2x x 2 4 x x 2 4 2x x x 2 4

48 2008 Fall Exam 19. Find the equation of the tangent line to the graph of y = lnx at the x point 1,0. a) y = x + 1 b) y = x + 1 c) y = x 1 d) y = lnx x 1

49 The End