Math 110 Final Exam General Review

Edward Yu

Da Game Plan

• Solving Limits – Regular limits – Indeterminate Form – Approach Infinities – One sided limits/Discontinuity

• Derivatives – Power Rule – Product/Quotient Rule – Chain Rule

• Marginal Functions – MR, MC, MP, Elasticity

• Implicit Differentiation & Related Rates

• Application of the 1st Derivative – Intervals of Increasing and Decreasing – Relative Extremas

• Application of the 2nd Derivative – Concavity – Inflection Points

• Law of Diminishing Returns

– The second derivative test

• Optimization – a.k.a Absolute Max/Min – Word Problems!!!

• 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)

Evaluating Limits

• Polynomials -> just substitute the limit in.

Ex: lim𝑥→−2

3𝑥2 + 𝑥 + 5

=3(−2)2+ −2 + 5 = 3 4 + 3 = 15

“Special case”

• Ex: lim𝑥→2𝑥2−4

𝑥−2

• =0

0

Indeterminate form

Indeterminate Form

Factoring

• lim𝑥→2𝑥2−4

𝑥−2

Rationalizing (Conjugate)

• lim𝑥→4

𝑥−4

𝑥−2

L'Hopital's Rule L'Hopital's Rule

Comparison

Factoring

• lim𝑥→2𝑥2−4

𝑥−2

=lim𝑥→2(𝑥+2)(𝑥−2)

𝑥−2

=lim𝑥→2 𝑥 + 2 = 4

Rationalization

• lim𝑥→4

𝑥−4

𝑥−2

=lim𝑥→4

𝑥−4

𝑥−2×

𝑥+2

𝑥+2

=lim𝑥→4

(𝑥−4)( 𝑥+2)

𝑥−4

=lim𝑥→4

𝑥 + 2

= 4

conjugate

𝑎 − 𝑏 𝑎 + 𝑏 = 𝑎2 + 𝑏2 𝑎 − 𝑏 𝑎 + 𝑏 = 𝑎2 + 𝑏2

Comparison

Factoring

• lim𝑥→2𝑥2−4

𝑥−2

=lim𝑥→2(𝑥+2)(𝑥−2)

𝑥−2

=lim𝑥→2 𝑥 + 2 = 4

L'Hopital's Rule

• lim𝑥→2𝑥2−4

𝑥−2

= lim𝑥→2

(𝑥2−4)′

(𝑥 − 2)′

=lim𝑥→22𝑥

1

=2(2) = 4

Limits approaching Infinites

• For Polynomials – Ex: lim

𝑥→∞5𝑥3 − 1

– ALWAYS DNE!!!

• For Rational Functions

– lim𝑥→±∞

𝑓(𝑥)

𝑔(𝑥)

– Three Cases 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑇𝑂𝑃 𝑖𝑠 𝐵𝑖𝑔𝑔𝑒𝑟

𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝐵𝑂𝑇𝑇𝑂𝑀 𝑖𝑠 𝐵𝑖𝑔𝑔𝑒𝑟𝑆𝑎𝑚𝑒

Degree of Top > Degree of Bottom

• Case 1: The highest power is in the numerator.

Always DNE (±∞).

• Ex: lim𝑥→∞

2𝑥5

3𝑥4 = 𝐷𝑁𝐸 𝑜𝑟 ∞

lim𝑥→∞

𝑥3

2 𝑥= 𝐷𝑁𝐸 𝑜𝑟 ∞

Degree of Bottom > Degree of Top

• Case 2: The highest power is in the denominator

Always = 0

• Ex: lim𝑥→∞

4𝑥−1

3𝑥4+2= 0

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: lim𝑥→∞

2𝑥3+7𝑥−4

5𝑥3−2𝑥=

2

5

Solving for Discontinuities

• Goal: Turn a discontinued function into differentiable everywhere.

• Key: Sub in the restriction of x!

• Ex: let 3𝑥 − 4 𝑖𝑓 𝑥 ≤ 2

𝑥2 + 𝑘𝑥 𝑖𝑓 𝑥 > 2 , what value of k

will make the function continuous everywhere?

• 3𝑥 − 4 = 𝑥2 + 𝑘𝑥,𝑤ℎ𝑖𝑙𝑒 𝑥 = 2 • 3 2 − 4 = 2 2 + 2𝑘 • 𝑘 = −1

Derivatives…

1. Power Rule 𝑥𝑛 2. Product Rule 𝑓(𝑥)𝑔(𝑥)

3. Quotient Rule 𝑓 𝑥

𝑔 𝑥

4. Chain Rule 𝑓 𝑥𝑛

Marginal Analysis

-It’s the change analysis via the 1st derivative!!

• C(x) -> Total Cost • R(x) -> Total Revenue

• P(x) -> Total Profit • Cost = Revenue – Profit • C(x) = R(x) – P(x) • R(x) = xf(x)

– f(x) is the price or demand function

• C’(x) -> Marginal Cost • R’(x) -> Marginal

Revenue • P’(x) -> Marginal

Profit

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)

Solution…

𝑅(𝑥) = 𝑝(𝑥) ∙ 𝑥 𝑅 𝑥 = 300 − 𝑥 𝑥 𝑅 𝑥 = 300𝑥 − 𝑥2 𝑅′ 𝑥 = 300 − 2𝑥 Since 50 units were sold.. 𝑅′ 50 = 300 − 2(50) 𝑅′ 50 = $200 −𝑡ℎ𝑒 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒 𝑟𝑒𝑣𝑒𝑛𝑢𝑒 𝑜𝑓 𝑠𝑒𝑙𝑙𝑖𝑛𝑔 𝑡ℎ𝑒 51𝑠𝑡 𝑢𝑛𝑖𝑡 𝑖𝑠 $200

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

Cause & Effect Causes

• Demand is Elastic

• Demand is Inelastic

• Demand is unitary

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 If E(p)=1 -> same amount!!

Example

• Staples has determined that the demand for erasers is given by: 𝑃 = 6 −

𝑥

20. P is the unit price in

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

• 𝑃 = 6 −𝑥

20

•𝑥

20= 6 − 𝑝

• 𝑥 = 120 − 20𝑝 = f(p)

• Remember 𝐸 𝑝 = −𝑃𝑓′(𝑝)

𝑓(𝑝)

• 𝑓′ 𝑥 = −20

• 𝐸 2 = −2 −20

120−20 2=

1

2

• Inelastic.

Implicit Differentiation

𝑦3 + 7𝑦 = 𝑥3

• Two-Step Problem to find 𝑑𝑦

𝑑𝑥𝑜𝑟 𝑦′.

• 1) Differentiate both sides in respect of x. When differentiating y, put a

𝑦’ or 𝑑𝑦

𝑑𝑥 after y

• 2) Solve the resulting function for

𝑦′or 𝑑𝑦

𝑑𝑥 in terms of x and y.

Let’s solve it!

• 𝑦3 + 7𝑦 = 𝑥3 • 3𝑦2𝑦′ + 7(1)𝑦′ = 3𝑥2 • 𝑦′ 3𝑦2 + 7 = 3𝑥2

• 𝑦′ =3𝑥2

3𝑦2+7

1st and 2nd Derivative Application

Local Extremas

Local Extremas

1st

Derivative

Increasing & Decreasing

Intervals

Increasing & Decreasing

Intervals

Slope Slope Tangent Tangent

1st and 2nd Derivative Application

2nd Derivative

Concavity Concavity Inflection

Points Inflection

Points

Point of Diminishing

Return

Point of Diminishing

Return

Approach: Increasing/Decreasing Intervals

1. Take derivative (finding slopes of the function) f’(x).

2. Set 𝑓′ 𝑥 = 0

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

Exercise

• 𝑓 𝑥 =2

3𝑥3 − 2𝑥2 − 6𝑥 − 2, where is f(x)

increasing/decreasing?

• 𝑓′ 𝑥 = 2𝑥2 − 4𝑥 − 6 • 0 = 2𝑥2 − 4𝑥 − 6 • 0 = 2 𝑥2 − 2𝑥 − 3 • 0 = 𝑥 − 3 𝑥 + 1

• 𝑥 = 3−1

Continued

• 𝑓′ 𝑥 = (𝑥 − 3)(𝑥 + 1) • 𝑓′ −2 = −2 − 3 −2 + 1 = 5 (positive) • 𝑓′ 0 = 0 − 3 0 + 1 = −3 (negative) • 𝑓′ 5 = 5 − 3 5 + 1 = 12 (positive) • Increase on −∞,−1 ∪ (3,∞) • Decrease on (−1,3)

+

-1 3

− +

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!

Concavities

– 𝑓′′ 𝑥 > 0, 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒 𝑖𝑠 𝑐𝑜𝑛𝑐𝑎𝑣𝑒 𝑢𝑝 – 𝑓′′ 𝑥 < 0, 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒 𝑖𝑠 𝑐𝑜𝑛𝑐𝑎𝑣𝑒 𝑑𝑜𝑤𝑛

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

Visually

In Economics In Math 110

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

Quick Exercise

• 𝑓 𝑥 =𝑥

𝑥2+1, over 0,2 Find absolute

max/min

• 𝑓′ 𝑥 =𝑥2+1−2𝑥(𝑥)

𝑥2+1 2

• 𝑓′ 𝑥 =1−𝑥2

𝑥2+1 2 = 0

• 𝑥 = 1; 𝑥 = −1

X=-1 not in the interval 0,2

• 𝑥 = 102

• 𝑓 𝑥 =𝑥

𝑥2+1

• 𝑓 1 =1

2

• 𝑓 0 = 0

• 𝑓 2 =2

5

Absolute Min

Absolute Max

Log/Ln Functions

• Since Log/Exponential Functions are inverses:

• 2 most important rules!

𝑒𝑙𝑛𝑥 = 𝑥 , 𝑥 > 0

𝑙𝑛𝑒𝑥 = 𝑥

Tricky Sample Exam Questions

• Solve for x:

• A) ln lnx = 1

• 𝑒ln 𝑙𝑛𝑥 = 𝑒1 • 𝑙𝑛𝑥 = 𝑒

• 𝑒𝑙𝑛𝑥 = 𝑒𝑒 • 𝑥 = 𝑒𝑒

Solve for x

• B) 𝑒𝑒𝑥= 10

• 𝑙𝑛𝑒𝑒𝑥= 𝑙𝑛10

• 𝑒𝑥 = 𝑙𝑛10 • 𝑙𝑛𝑒𝑥 = ln 𝑙𝑛10 • 𝑥 = ln 𝑙𝑛10

Interests Problems

• General Compound Interest

𝐴 = 𝑃 1 +𝑟

𝑚

𝑚𝑡

• Continuous Compound Interest 𝐴 = 𝑃𝑒𝑟𝑡

• Effective Interest Rate (EFF)

𝑟𝑒𝑓𝑓 = 1 +𝑟

𝑚

𝑚

− 1

• Present Values – General compounded (periodic) – Continuous compounded (e)

Effective Interest Rates Ex.

• Find the effective rate of interest corresponding to: a) 10%/year compounded semi-annually b) 9%/year compounded quarterly

a) 𝑟𝑒𝑓𝑓 = 1 +0.10

2

2− 1 = 10.25%/𝑦𝑒𝑎𝑟

b) 𝑟𝑒𝑓𝑓 = 1 +0.90

4

4− 1 = 9.308%/𝑦𝑒𝑎𝑟

Summing up…

• 𝐴 = 𝑃 1 +𝑟

𝑚

𝑚𝑡

• 𝐴 = 𝑃𝑒𝑟𝑡

• 𝑟𝑒𝑓𝑓 = 1 +𝑟

𝑚

𝑚− 1

• 𝑃 = 𝐴 1 +𝑟

𝑚

−𝑚𝑡 (Present Value)

• 𝑃 = 𝐴𝑒−𝑟𝑡 (Present Value)

TIME TO GET REAL

2008 Fall Exam

• 13. Find the derivative of 𝑓 𝑡 = 𝑒2𝑡ln (𝑡 + 1).

• a) 2𝑒2𝑡

𝑡+1

• b) 𝑒2𝑡

𝑡+1+ 2𝑒2𝑡 ln 𝑡 + 1

• c) 2𝑒2𝑡 +1

𝑡+1

• d) 𝑒2𝑡

𝑡+1+ 𝑒2𝑡ln (𝑡 + 1)

2008 Fall Exam

• 10. Simplify ln 2𝑒5𝑥 + ln𝑒3𝑥

2

• a)8𝑥 • b)𝑒8𝑥

• c)𝑙𝑛2 − ln1

2+ 8𝑥

• d) 𝑒5𝑥 2 + 𝑒3𝑥1

2

2008 Fall Exam

• 14. Find the derivative of

𝑓 𝑥 = 𝑙𝑛 𝑥2 − 4

a)2𝑥

𝑥2−4

b)𝑥

𝑥2−4

c)2𝑥

𝑥2−4

d)1

𝑥2−4

2008 Fall Exam

• 19. Find the equation of the tangent line to the graph of 𝑦 =

𝑙𝑛𝑥

𝑥 at the

point 1,0 .

a) 𝑦 = 𝑥 + 1 b) 𝑦 = −𝑥 + 1 c) 𝑦 = 𝑥 − 1

d) 𝑦 =𝑙𝑛𝑥

𝑥− 1

The End