mat1300 final review - university of...

MAT1300 Final Review

Pieter Hofstra

December 4, 2009

MaterialTopicsFormat

Sample Problems

Sections from the book to study (8th Edition)

Chapter 0:

0.1: Real line and Order0.2: Absolute Value and Distance0.3: Exponents and Radicals0.4: Factoring Polynomials (you may omit the part on therational zero theorem, pp. 23)0.5: Fractions and Rationalization

Pieter Hofstra MAT1300 Final Review


Sample Problems


Chapter 1:

1.1: Cartesian Plane and Distance (you may skip the part ontranslating points, p. 39)1.2: Graphs of Equations1.3: Lines and Slope (may skip linear depreciation)1.4: Functions1.5: Limits1.6: Continuity (may skip applications)



Sample Problems


Chapter 2:

2.1: Derivative and Slope of a Graph (you may skip the parton translating points, p. 39)2.2: Rules of Differentiation2.3: Rates of Change: velocity and marginals2.4: Product and Quotient Rules2.5: Chain Rule2.7: Implicit Differentation2.8: Related Rates



Sample Problems


Chapter 3:

3.1: Increasing and Decreasing Functions3.2: Extrema and the First Derivative Test3.3: Concavity and the Second Derivative Test (may skipdiminishing returns)3.4: Optimization Problems3.5: Business and Economics Applications3.6: Asymptotes



Sample Problems


Chapter 4:

4.1: Exponential Functions4.2: Natural Exponential Functions (may skip Example 4)4.3: Derivatives of Exponential Functions4.4: Logarithmic Functions4.5: Derivatives of Logarithmic Functions4.6: Exponential Growth and Decay



Sample Problems


Chapter 5:

5.1 Antiderivatives and Indefinite Integrals5.2 Substitution and General Power Rule (skip p. 371)5.3 Exponential and Logarithmic Integrals5.4 Area and the Fundamental Theorem (up to p. 387)5.5 Area Bounded by Two Graphs



Sample Problems


Chapter 6:

6.1 Integration by Parts and Present Value6.5 Improper Integrals (skip the vertical asymptote examples)



Sample Problems

Chapter 7:

7.1 The 3d Coordinate System7.2 Surfaces in Space (only first 2 pages)7.3 Functions of Several Variables (we only do 2 variables, upto p. 499)7.4 Partial Derivatives (may skip pp. 509-510)7.5 Extrema of Functions of Two Variables (up to p. 520)



Sample Problems

Key concepts from Chapter 1:

1 Graphing equations; finding intercepts

2 Break-even analysis; finding a break-even point

3 Slope of a line; finding equation y = mx + b for a line

4 Functions: domain, range; finding domain and range of afunction

5 Composite and inverse: finding the inverse of a function,horizontal line test

6 Limits: one- and two-sided; evaluating the limit

7 Operations on limits; calculating limits of polynomials andrational functions

8 Continuity; understand definition and apply to given functions



Sample Problems


1 Definition of derivative: Use definition to calculate derivativeof standard functions

2 Interpretation of derivative as slope of tangent line: findingequation of tangent line, demand functions

3 Differentiability: understanding why certain functions are notdifferentiable at some points

4 Rules for differentiation: be able to use all the rules (andcombine them as needed)

5 Implicit differentiation: finding derivative of implicitly definedfunctions

6 Related Rates; Identifying dependent and independentvariables, using the chain rule, solving for unknown rate.



Sample Problems


1 Increasing/decreasing behaviour of a function; using thederivative to find where a function is increasing or decreasing

2 Critical points of a function; finding critical points

3 Local extrema; classifying the critical points using first- orsecond derivative test

4 Absolute extrema; finding the absolute min or max of afunction on a given interval

5 Optimization; solving optimization problems, both geometrical(volume, area, etc.) and economical (cost, revenue, profit)

6 Elasticity: computing price elasticity of demand; elastic vs.inelastic

7 Asymptotes; finding horizontal and vertical asymptotes ofrational functions



Sample Problems


1 Exponential functions; natural base e; know graphs,calculation rules

2 Compound interest: know formulas for compound interest andcontinuous compounding and know how to apply these

3 Logarithms; know graphs, calculation rules; apply logarithmsto solve exponential equations, e.g. in compound interestproblems

4 Derivatives of exponential functions; know calculation rules

5 Derivatives of logarithms: know calculation rules

6 Exponential growth and decay; setting up a formula, usinggiven data to solve for unknowns; solve problems similar tothe examples studied in class (population growth, compoundinterest, exponential decay



Sample Problems


1 Antiderivatives; general versus particular solutions; findingantiderivatives of polynomial functions; finding a particularsolution using initial data

2 Power Rule and Substitution: using these rules to solveintegrals; rewriting integrals so that these rules becomeapplicable

3 Exponential and logarithmic integrals; basic integration rules;combining these with substitution rule

4 FTC, Area under graph; finding area under a graph of afunction; difference between area and definite integral

5 Area between two curves; finding intersection points; locatingarea; calculating area using FTC; consumer and producersurplus



Sample Problems


1 Integration by parts; using formula for IBP, recognizing whento use this (as opposed to, say substitution); present value

2 Improper Integrals; definition using limit; divergence vs.convergence; evaluating improper integrals; present value



Sample Problems


1 3d coordinates; x,y,z axis, coordinate planes, general planes

2 Surfaces; skim this section, it helps you understand the rest

3 Functions; evaluating functions; finding the domain of afunction; level curves

4 Partial Derivatives; finding partial derivatives usingdifferentiation; geometric interpretation; second partialderivatives

5 Extrema; critical points; finding critical points by solving twoequations simultaneously; using second derivative test toclassify critical points



Sample Problems

Format of the exam:

The final exam consists of:

10 MC questions (worth 30% in total)

4 on derivatives and applications4 on integrals and applications2 on functions of 2 variables

5 long answer questions (worth 70% in total)

one related rates problemone area problemone indefinite integral problem

The exam is 3 hours long. No books, notes or calculators areallowed.



Sample Problems

Problem 1

Suppose that for a certain product, the demand function is givenby D(x) = 11− x2 and the supply function is given byS(x) = 2x + 3. Calculate the producer surplus.

Solution. Find the equilibrium price:

D(x) = S(x) ⇔ 2x + 3 = 11− x2

⇔ x2 + 2x − 8 = 0

⇔ (x − 2)(x + 4) = 0

⇒ x = 2, x = −4

We only consider x = 2. The equilibrium price is p = 7.



Sample Problems

Problem 1

CS =

∫ 2

011− x2 − 7dx =

∫ 2

04− x2dx = 4x − x3

3|20

= 8− 8

3=

16

3

PS =

∫ 2

07− (2x + 3)dx =

∫ 2

04− 2xdx = 4x − x2|20

= 8− 4 = 4



Sample Problems

Problem 2

If f (x) is a function such that f ′(x) = e2x and f (ln(3)) = 5, findf (0).

Solution. We have f (x) =∫

e2xdx = 12e2x + C .Use the initial

data f (ln(3)) = 5 to solve for C :

5 =1

2e2 ln(3) + C =

1

2· 32 + C =

9

2+ C ⇔ C =

1

2

Thus

f (x) =1

2e2x +

1

2



Sample Problems

Problem 2



e2xdx = 12e2x + C .

Use the initialdata f (ln(3)) = 5 to solve for C :

5 =1

2e2 ln(3) + C =

1

2· 32 + C =

9

2+ C ⇔ C =

1

2

Thus

f (x) =1

2e2x +

1

2



Sample Problems

Problem 2



e2xdx = 12e2x + C .Use the initial

data f (ln(3)) = 5 to solve for C :

5 =1

2e2 ln(3) + C =

1

2· 32 + C =

9

2+ C ⇔ C =

1

2

Thus

f (x) =1

2e2x +

1

2



Sample Problems

Problem 3.

Calculate:∫∞1

dx3√x

.

Solution. We have∫ ∞1

dx3√

x= lim

b→∞

∫ b

1x−1/3dx

= limb→∞

[3

2x2/3|b1]

= limb→∞

[3

2b2/3 − (−3

2· 1)]

= ∞

The limit does not exist, so the integral diverges.



Sample Problems

Problem 4.

Consider the function of two variablesf (x , y) = 2x2 − 4xy + 4y2 − 5x + 9y + 3.

Calculate the first-order partial derivatives.Find all critical points.Identify what type of critical points they are (local max, localmin or saddle point).

Solution. For the derivatives, calculate

fx = 4x − 4y − 5, fy = −4x + 8y + 9

To find critical points, we need fx = 0 and fy = 0. Add theequations to get

4y + 4 = 0⇒ y = −1, x =1

4

Thus there is one CP, namely (14 ,−1).



Sample Problems

Problem 4.

We calculate the second derivatives:

fxx = 4, fyy = 8, fxy = −4

Then

D = fxx fyy − f 2xy = 4 · 8− (−4)2 = 32− 16 = 16 > 0

Since D > 0 and fxx > 0, we have a local min at (14 ,−1).



Sample Problems

Problem 5.

The profit function of a company is given by P(x) = 2x2 − 200x .The current production level is 100 units, and production isdecreasing by 2 units per day. At what rate is profit decreasing?

Solution. This is a related rates problem. Identify the threevariables: P, x , t.Since x depends on t and P depends on x wehave

dP

dt=

dP

dx

dx

dt

We know dxdt = −2. We need to find dP

dt . So we must calculate dPdx :

dP

dx= 4x − 200.

At x = 100, this gives dPdx = 200.Thus dP

dt = 200 · (−2) = −400.Therefore profit is decreasing by $400 per day.



Sample Problems

Problem 5.


Solution. This is a related rates problem. Identify the threevariables: P, x , t.

Since x depends on t and P depends on x wehave

dP

dt=

dP

dx

dx

dt



dP

dx= 4x − 200.





Sample Problems

Problem 5.



dP

dt=

dP

dx

dx

dt



dP

dx= 4x − 200.





Sample Problems

Problem 5.



dP

dt=

dP

dx

dx

dt



dP

dx= 4x − 200.

At x = 100, this gives dPdx = 200.

Thus dPdt = 200 · (−2) = −400.

Therefore profit is decreasing by $400 per day.



Sample Problems

Problem 5.



dP

dt=

dP

dx

dx

dt



dP

dx= 4x − 200.





Sample Problems

Problem 6.

Consider f (x , y) =√−xy . Find the domain of f .

Solution. For the domain, note that we must have anon-negative value in the square root. Thus we must have−xy ≥ 0, i.e. xy ≤ 0. This means that either x is positive and y isnegative or vice versa. Thus the domain is

dom(f ) = {(x , y)|x ≤ 0&y ≥ 0} ∪ {(x , y)|x ≥ 0&y ≤ 0}

(This region corresponds to the union of the second and fourthquadrants in the plane.)


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