Math 120 Ex 2 Study Guide PDF

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MATH 120 EX 2 STUDY GUIDE CHAPTERS 2 AND 3 AND BARNETT CH 3

You should be able to: Chapter 2 Apply all derivative formulas (power rule, product rule, quotient rule, chain rule, etc from Chapter 1) Find intervals of increase (f’(x) > 0) and decrease (f’(x) < 0) Find and classify critical values (where f’(x) = 0, undefined, or does not exist) using the first derivative test Sketch the graph using techniques of algebra (finding x & y intercepts) and calculus (finding intervals of increase and decrease, critical values, and end behavior) Ex 1: Find and classify the critical values and sketch the graphs of the following functions: a) b)

−1 3 x +3 x2 −9 x+2 3 2/ 3 g ( x) =1−x f ( x) =

Ex 2: Draw a graph to match the description given f’(x) < 0 over

(−∞ , 2)∧ ( 5,9 ) and f’(x) > 0 over

(2,5 )∧(9 , ∞ )

Find intervals of upward concavity ( f’’(x) > 0) and downward concavity ( f’’(x) < 0) Find inflection points. If f changes concavity where f’’(x) = 0 or is undefined (say, at x = a), then f has an inflection point where x = a. If f does not change concavity where f’’(x) is undefined, then f has a cusp at x = a. Use the Second Derivative Test to classify critical values (If c is a critical value of a function f and f’’(c) > 0, then f has a relative minimum where x = c, if f’’(c) < 0, then f has a relative maximum where x = c) Ex: Find intervals of upward/downward concavity and inflection points and/or cusp of the Examples 1 and 2. Ex 3: Use the Second Derivative Test to classify the critical value(s) of

f ( x )=x 4 −2 x 3

Ex 4: Sketch the graph in Example 3 finding intercepts, critical values, inflection points, end behavior, etc. Graph rational functions: Be able to include the concept of horizontal and vertical asymptotes and open holes. Remember, you have to determine the domain of f(x) FIRST. Then,

reduce f(x) to its simplest terms. Any values that make the reduced rational function’s denominator zero are where f has a vertical asymptote. If a value makes the denominator of f zero but NOT the reduced rational function’s denominator zero then f

If lim f (x ) =L and L is x →± ∞

has an open hole at that value. For horizontal asymptotes:

finite, then y=L is a horizontal asymptote (note: you have to test both directions). If L is infinite, then f does not have a horizontal asymptote. Find Absolute Extrema over a closed interval (you need to include the values of the function at the endpoints of the closed interval as well as at critical values) Ex 5: Use function from Example 3. Find the Absolute Max and Abs Min over the interval [-2,1] Solve Max-Min Problems: Work some of the exercises in 2.5 (especially from #23-51). Try 52 and 53 too. Try odd # so you can check your answers. Find differentials (just remember, it’s dy = f ’ (x)*dx). If you’re given a value of x and dx, you can find dy. If x and dy, you can find dx. Ex 6: For

2 y=f ( x ) = x+ x , x =3∧ ∆ x =dx =.01, find dy

Marginals (Profit, Revenue, Cost): change in profit, revenue, or cost by increasing x to x+1. This is just f’(x) for whatever value of x you’re given. Ex 7: For

C ( x ) =0.002 x 3+ 0.1 x 2 +42 x+ 300

Currently: 40 items are produced daily and C is in thousands.Determine the following: a) What is the current daily cost b) What would be the additional daily cost of increasing daily production to 41 items? c) What is the marginal cost when x = 40? d) Use marginal cost to estimate the daily cost of increasing production to 42 items daily.

unit

Find, interpret the output of, and find price to maximize revenue using elasticity of demand. You will be given the formula. Revenue is maximized when E(x) = 1 (demand is elastic). For price x and demand D (x) , elasticity of demand is given by:

|

E ( x) =

xD ' (x) D (x )

|

E ( x ) < 0→ raising price increases revenue, lowering revenue elastic: raising price reduces revenue, lowering revenue Demand is unit elastic: raising or lowering price has little impact on revenue. This is where revenue is maximized

Demand is inelastic: price decreases E ( x ) > 0→ Demand is price increases

E ( x ) =1→

Chapter 3 Know the derivatives of

a x , e x , log a x , ln x and be able to find the derivatives if a g( x) , e g (x) , log a g (x), ln g (x)

You will be given all finance formulas but not told what each does or what each variable means. You will NOT be given the finance flow chart currently found in course documents. You should be able to apply these formulas whether the application is finance (interest) or life science (like population). Know doubling and half-life. Remember, for multiplying problems, the initial amount does not affect the doubling or halving time. If you are trying to double or halve a quantity defined by a finite number of compounding periods (like annual, semi-annual, quarterly, etc), you use the formula for compounded interest. If you are doing a population or optimum doubling time problem or if “continuously compounded” is

P (t )=P e rt

mentioned, you use doubling or halving time. If negative, it’s halving.

r is

where

r=

ln 2 where T T

is the

positive, the it’s doubling, if it is

DO NOT FORGET YOUR APPROVED CALCULATOR!!!!!!! PLEASE HAVE YOUR RED ID ON YOUR SCANTRON PROPERLY AND COMPLETELY FILLED OUT AND BUBBLED IN !!!!...