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2015 - 2016

CalculusThis course is designed to provide a firm background and understanding of the basic concepts of calculus; including limits, differentiation, applications of derivatives, exponential/logarithmic functions, integration, applications of integration and trigonometric functions.

Course Information:

Frequency & Duration: Daily for 42 minutesText: Barnett, Raymond, et al. Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Upper Saddle River: Prentice Hall, 2002

Calculus v. 2015 - 2016

Content: Functions Duration: Aug. /Sept (2 weeks)

Essential Question:

What information needs to be known / found in order to graph a function?What strategies can be used to solve for unknowns in algebraic equations?

Skill:

Graph all elementary functions and transformations of these functions. Summarize the key components of a parent, polynomial, rational,

exponential, and logarithmic function’s graph (including intercepts, asymptotes, end behavior, and turning points/extrema).

Graph a quadratic (standard form or vertex form), rational, polynomial, exponential, and logarithmic function using the key components and/or transformations.

Solve a polynomial equation by factoring (including perfect square, difference of squares, sum of cubes and difference of cubes), quadratic formula and factoring by grouping.

Solve a rational, radical, exponential, and logarithmic equation.

Assessment:

Graph y=−2 3√ x+3−1 Graph y=e−x+3 Graph y=ln ( x+3 )−4

Analyze and graph f ( x )= −3 x( x+3 )2

Analyze and graph f ( x )=2x4−5x3−4 x2+3 x+6 Solve: x2−5 x=6 Solve: 5 (3 x−7 )3+x (3 x−7 )2=0

Solve: 2x=5+1x2+ x

Solve: 3+√3 x+1=x

Resources: Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Chapter 1 & 2 (pages 3-125, 596-647)

Standards: This content is beyond the scope of the PA Core Mathematical Standards.

Vocabulary:

Exponential Function – Any function in the form: f ( x )=bx, where b is any real number such that b>0 aand b≠1; Logarithmic Function – Any function in the form: f ( x )=logb x, where b is any real number such that b>0 aand b≠1; ¿Polynomial Function– An function in the form

Calculus v. 2015 - 2016

f ( x )=an xn+an−1 x

n−1…a2 x2+a1 x+a0. Where n is an integer and a is a number;

Quadratic Function– A function in the form f ( x )=a x2+bx+c or f ( x )=a ( x−h )2+k ;Radical Function– A function that contains a radical expression in the radicand; Rational Function – An algebraic fraction such that the numerator and denominator are polynomials

Comments:

Content: Evaluating Limits Duration: Sept./Oct. (4 Weeks)

Essential Question:

What is the limit of a function and how can the limit of a function be computed? (What is the best/proper method in order to determine this?)How can limits be used to explain the asymptotes and continuity of a function?

Skill:

Define and understand the limit of a function. Determine the one sided limits of a function. Evaluate the limit of a function at any point on the graph of a function

(Graphically, tables, and algebraically). Determine when a limit does and does not exist. Evaluate infinite limits and limits at infinity. Construct a possible graph for a function given different limits. Understand the definition of continuity in terms of limits

Assessment:

Define limit. Use a graph to determine the value of a limit. Make a table of values to determine the value of the limit. Evaluate a limit using algebraic techniques (simplification, cancellation,

rationalization and multiplying by conjugates). Determine the limit at a point of discontinuity (holes, vertical asymptotes

and breaks). Determine the end behavior of the graph (limit at infinity) Given the limit values of a function create a possible graph for the function.

Where is the function, ( x )= x−35−x

, discontinuous? Use the limit definition to explain why.

Resources:Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 3-2 (pages 144-161)Calculus: AP Edition. Pearson: Chapter 2 (pages 61-119)Graphing Calculator


Vocabulary:

Continuity– A function is continuous at a point if the following three conditions

Calculus v. 2015 - 2016

are met: 1. limx→af ( x ) exists; 2. f (a ) exists; 3. lim

x→af ( x )=f (a ); Indeterminate

Form– When direct substituting to evaluate a limit the result is 00

. This means that the limit cannot be determined with the function in this form. More must be done to evaluate the limit; Infinite Limits– A limit that is equal to ±∞. The function grows without bound as x approaches a given number; Limit– The value that y “approaches” as x approaches a given number; Limits at Infinity– The value that y “approaches” as x approaches ±∞; One-Sided Limit– The value that y “approaches” as x approaches a given number form one side of the x-value

Comments:

Content: Rates of Change/Derivative Definition

Duration: October (3 Weeks)

Essential Question:

What is the difference between Average Rate of Change and Instantaneous Rate of Change?How do you calculate the slope of a non-linear function?

Skill:

Graphically interpret the average rate of change as the slope of a secant line.

Determine the average rate of change. Graphically interpret the instantaneous rate of change as the slope of a

tangent line or slope of the graph. Determine the instantaneous rate of change graphically and algebraically

using the four step process. Define a derivative and know the formula for a derivative. Calculate a derivative using the definition and the four step process.

Assessment:

Given a function and two x-values calculate the average rate of change between the two points.

Relate the slope of a secant line to the difference quotient formula. Given a function, calculate the instantaneous rate of change at a given x-

value. Estimate the instantaneous rate of change from a graph. Know and explain the formula for finding the instantaneous rate of change.

o f ( x )=2x+3 at x=−4o t ( x )=3−7√ x at x=−1

Know and explain the formula for a derivative. Given a function calculate the derivative and find the slope at a given point.

o Find y ' given y=1−x2

o Find k ' ( x ) given k ( x )= 12−x

o Find g' (x ) given g ( x )=24

Resources:Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 3-2 (pages 131-144 & 162-175)Calculus: AP Edition. Pearson: Chapter 2 (pages 136-153)

Calculus v. 2015 - 2016


Vocabulary:

Average Rate of Change– Slope of a secant line (m=f (a )−f (b )a−b ); Derivative–

A function that gives the slope of a tangent line at any point in the domain of

the function ( limh→0 f ( x+h )− f (x )h ); Difference Quotient– ( f ( x+h )−f ( x )

h ); Instantaneous Rate of Change– Slope of a tangent line ( limh→0 f (a+h )−f (a )

h )Comments:

Content: Rules for Differentiation Duration: November (4 Weeks)

Essential Question:

How do you find the slope of the tangent line at any point on the graph of the function?

Skill:

Know and understand the rules for differentiation. Including constant, constant times a function, power, sum and difference, product, quotient, chain, logarithmic and exponential rules.

Evaluate the derivative of a function using the rules for differentiation.

Use implicit differentiation to calculate dydx

. Determine when a function is non-differentiable (graphically and

algebraically). Write the equation of the tangent line at a point on the graph of a function. Determine where a function has a horizontal tangent line (graphically and

algebraically).

Assessment: Find the derivative of f ( x )=( 32x + x3 )

Find the derivative of f ( x )=( ln x+e−x ) Find the derivative of y=log (2−x ) Find the derivative of f ( x )=32x+ x

2

Find the derivative of y=√ ln(x2−2 x ) Find the derivative of f ( x )= ex+5

ln x2

Find dydx

if x=e y+7 Given a function f ( x ), find the x values for which the function is non-

differentiable and explain why the derivative does not exist at these points.

Calculus v. 2015 - 2016

Given a function and an x value find the equation of a tangent line at that x value.

Write the equation of the tangent line at x=8 for the function

f ( x )=3 x23−5 3√ x

Write the equation of the tangent line at x=e for the function f ( x )=ln x2

Given a function, algebraically find where it has a horizontal tangent line. ( f ' ( x )=0)

f ( x )= x( x+3 )4

g ( x )= xln x

Given the graph of a function, determine where f ' ( x )=0 and where f ' ( x ) does not exist.

Resources:Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections 3.4, 3.5, 3.6, 5.2, 5.3 & 5.4 (pages 175 – 204 & 322 – 343 & 345 – 351)Calculus: AP Edition. Pearson: Sections 3.3, 3.4 & 3.7- 3.9 (pages 153 – 221)


Vocabulary:

Differentiable– The derivative of the function exists at a given point(s); Differentiate– Find the derivative of the function; Differentiation– The process of finding a derivative; Explicit Equation– An equation that defines the dependent variable ( y ) in terms of the independent variable( x );Implicit Equation– An equation that does not define the dependent variable ( y ) in terms of the independent variable ( x ); Non-differentiable– The derivative of the function does not exist at a given point(s). A function is non-differentiable at any point where the graph is discontinuous, has a sharp point, has a vertical tangent line or at the end points.

Comments:

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Content: Derivative Applications Duration: December (3 Weeks)

Essential Question:

How can a derivative be applied to a real world situation (business, economics, science, etc)?

Skill: Differentiate an equation with respect to time ( t ). Use differentiation to solve related rate problems. Relate distance, velocity, and acceleration using differentiation. Use derivatives to analyze business equations.

Assessment:

Given a function, x value, and dxdt

. Find dydt

A construction worker pulls a 16-foot plank up the side of a building under construction by means of a rope tied to the end of the plank. If the worker pulls the rope at a rate of 0.5 ft/sec. How fast is the end of the plank sliding along the ground when it is 8 feet from the wall?

Given the distance equation, find the velocity equation and acceleration equation.

Interpret the slope of the distance and velocity function. Given the price and cost equations, find the revenue and profit equations. Find the marginal cost, marginal revenue, marginal profit, average cost,

average revenue, average profit, marginal average cost, marginal average revenue and marginal average profit, given the price and cost equation.

Interpret the result from evaluating the derivative of a business function. Find R' (500 ) and interpret. Find C ' (1000 ) and interpret. Find P (200 )and interpret.

Resources:

Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections 3.4, 3.7 & 5.5, (pages 181 – 188 & 352 – 360)Calculus: AP Edition. Pearson: Sections 3.6, 3.11 (pages 181 – 191 & 232 – 240)

ACTIVITY: Tootsie Pop Lab from A Watched Cup Never Cools from Key Curriculum Press.


Vocabular Average (Cost, Revenue, Profit)– The average cost, revenue or profit per

Calculus v. 2015 - 2016

y:item; Marginal Average (Cost, Revenue, Profit)– The rate at which the average cost, average revenue or average profit is changing; Marginal (Cost, Revenue, Profit)– The rate at which the cost, revenue or profit is changing. Or, the approximate cost, revenue, or profit of the next item

Comments:

Calculus v. 2015 - 2016

Content: Graphical Derivative Applications

Duration: January (4 Weeks)

Essential Question:

How can the concept of a derivative be used to interpret/explain the graph of a function?

Skill:

Determine where a function is increasing or decreasing. Find the local extrema occur of a function using the first and second

derivative test. Determine the concavity and inflection points of a function. Draw an accurate graph of a function and its derivative.

Assessment:

Use the relationship between f ( x ) and f ' ( x ) to answer graphical questions given f ( x ), and f ' ( x ).

o Given a graph of f ( x ) determine where f ' ( x ) is negative.o Given a graph of f ' ( x ) determine where f ( x ) increasing.o Given a graph of f ' ( x ) determine where f ( x ) has a local minimum.

Use the relationship between f ( x ), f ' ( x ), and f ' ' ( x ) to answer graphical questions given f ( x ), f ' ( x ), and f ' ' ( x ).

o Given a graph of f ( x ) determine where f ' ( x ) is decreasing.o Given a graph of f ' ( x ) determine where f ' ' ( x ) positive.o Given a graph of f ' ' ( x )determine where does f ( x ) have inflection

points. Given an equation for f ( x ) determine where the function is increasing and

decreasing by making a sign chart for f ' ( x ).

o f ( x )= 3x2−9

o f ( x )=x e−x

Given an equation for f ( x ) use the first derivative test to interpret the sign for f ' ( x ) and determine where the function has local extrema.

o f ( x )=x23−x

53

o f ( x )= ln xx

, x>0 Given an equation for f ( x ) determine where the function is concave up,

concave down and where inflection points occur by making a sign chart for f ' ' ( x ).

o f ( x )= 3√( x−3 )+1o f ( x )=ln (x2−2x+10 )

Given an equation for f ( x ) use the second derivative test to determine where the function has local extrema.

o f ( x )= (x−2 ) (3−x )5

Calculus v. 2015 - 2016

o f ( x )=4+x+ 9x

Given a sign chart for f ' ( x ) and f ' ' ( x ), construct an accurate graph of f ( x ) and f ' ( x ).

Given the graph of f ' ( x ), construct a sign chart for f ' ( x ) and f ' ' ( x ) then construct an accurate graph for f ( x ).

Given the graph of f ( x ), construct a sign chart for f ' ( x ) and f ' ' ( x ) the construct an accurate graph for f ' ( x ).

Resources:Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections 4.2 & 4.3, (pages 242 – 274)Calculus: AP Edition. Pearson: Sections 4.1 & 4.2 (pages 246 – 271)


Vocabulary:

Concave up– When the graph of a function opens up. This occurs when f ' ' ( x ) is positive or when f ' ( x ) is increasing; Concave down– When the graph of a function opens down. This occurs when f ' ' ( x ) is negative or when f ' ( x ) is decreasing; Critical values– Partition numbers for f ' ( x ) that are in the domain of f ( x ); Decreasing– When the slope (or derivative) of a function is negative; First derivative test– A method to find the local extrema of a function by interpreting the sign chart for f ' ( x ); Increasing– When the slope (or derivative) of a function is positive; Inflection point– A point on the graph of a function where the concavity changes. At this point, f ' ' ( x )=0 and f ' ( x ) has local extrema; Local extrema– Where the graph of a function has a local minimum or local maximum. These occur when the derivative changes form positive to negative or vice versa; Partition numbers ( for f ' ( x ) )– Points where the function ( f (x ) ) is equal to zero or discontinuous; Second derivative test– A method to find the local extrema of a function by using the second derivative to analyze the critical values for f ( x )

Comments:

Calculus v. 2015 - 2016

Content: Optimization Duration: February (2 Weeks)

Essential Question:

What techniques can be used to find the absolute maximum or minimum value of a function?How can a real world problem be solved using absolute extrema?

Skill:

Understand the extreme value theorem. Find the absolute extrema for a given function on a closed interval and open

interval. Setup and solve real world optimization problems using the proper steps.

1. Find the primary equation.2. Find a secondary equation (if needed)3. Write the primary equation in terms of one variable.4. Use the proper techniques of calculus to find the desired maximum or

minimum value.5. Answer the question.

Assessment:

Find the absolute extrema for the function f ( x )=x2 (3−ln x )on the interval (0 ,∞ ).

Find the absolute extrema for the function f ( x )=√x2+x+12 on the interval [−2,5 ]

Find two positive numbers such that the sum of the first and three times the second is 38 and the product is maximized.

A Norman window has the shape of a rectangle topped with a semi-circle. (Thus the diameter of the semi-circle is equal to the width of the rectangle.) If the perimeter of the window is 30 feet, find the dimensions (radius and height) of the window so that the greatest possible amount of light is admitted.

A man is at point A on the bank of a straight river, 3 km wide, and wants to reach point B, 8 km downstream on the opposite bank, as quickly as possible. He could row to point C and then run to point B, or he could row directly to point B, or he could row to some point D between C and B and then run to point B. If he can row at 6 km/h and run at 8 km/h, where should he land to reach be as soon as possible?

A rectangular storage container with an open top is to have a volume of 10 m3. The length of the base is twice the width. Material for the base cost $ 10 per square meter. Material for the sides costs $ 6 per square meter. Find the cost of the materials for the cheapest such container.

Given the following cost function; C(x) = 0.001x3 – 5x + 250; where the cost is in dollars and x is the number of units produced, find the minimum average cost per unit.

Resources:

Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 4.5 (pages 292 – 308)Calculus: AP Edition. Pearson: Sections 4.4 (pages 281 – 291)

ACTIVITY: Pop Can Optimization (based on # 45 on page 288 of AP Edition. Pearson)

Calculus v. 2015 - 2016


Vocabulary:

Absolute maximum– The highest point on the graph of a function. There is an absolute maximum a x=c if f ( c )≥ f ( x ) for all x; Absolute minimum– The lowest point on the graph of a function. There is an absolute minimum a x=c if f ( c )≤ f ( x ) for all x; Extreme value theorem– If a function is continuous on an interval, the function will have an absolute maximum and absolute minimum on that interval; Primary equation– The equation that needs to maximized or minimized.

Comments:

Calculus v. 2015 - 2016

Content: Anti-differentiation Duration: February/March (3 Weeks)

Essential Question:

If you know the derivative of a function, how can you work backwards to find the original function?

Skill:

Understand the relationship between the derivative and the anti-derivative. Know the difference between the antiderivative and indefinite integral. Recognize that there are an infinite number of anti-derivatives for a

function and graphically interpret this idea. Know and understand the rules for integration.

o ∫ k dx=kx+c when k is a constant

o ∫ xndx= xn+1

n+1+c except when n=−1

o ∫ exdx=ex+co ∫ 1x dx=ln|x|+co ∫ kf (x )dx=k∫ f ( x )dxo ∫ f ( x )dx ±∫ g ( x )dx=∫ [ f ( x )± g ( x ) ]dx

Find the antiderivative of a function using the proper rules. Understand the differential as a place holder for use with integration by

substitution. Understand the process of integration by substitution. Use integration by substitution to evaluate an indefinite integral. Know when it is appropriate to use integration by substitution.

Assessment:

Define Anti-differentiation and Integration. Answer questions based on Integration properties and definitions..

o True/False: A constant factor can be moved across the integral sign.o True/False: The integral of a constant function is zero.o True/False: A function has only one anti-derivative.

Given the graph of one anti-derivative, find a possible graph for another anti-derivative of the same function.

Given the graph of a function, find a possible graph for an anti-derivative. Find a particular anti-derivative, given f ( x ) and an initial condition. Given f ( x )=2x+ln x, find the differential dy .

Evaluate: ∫( 3x +5 x4

3 )dx Evaluate: ∫ (4e x+xe−ex )dx

Evaluate: ∫ x3+2x−3x

dx

Evaluate: ∫ 3 xx2+5

dx

Evaluate: ∫ 1+e− x

exdx

Evaluate: ∫ x2 e3 x2

dx

Calculus v. 2015 - 2016

Evaluate: ∫ ln xx dx Evaluate: ∫ x (3−x )5dx

Resources:

Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections 6.1 & 6.2 (pages 363 – 389)Calculus: AP Edition. Pearson: Sections 5.1 & 5.6 (pages 337 – 347 & 401 – 211)


Vocabulary:

Anti-derivative – The reconstruction of a function from its derivative; Differential – dy=f ' ( x)dx ; Integral – The family of all anti-derivatives; Integral Symbol – ∫❑; Integrand – The function that is being integrated. f ( x ) in the integral ∫ f ( x )dx ; Variable of Integration – The independent variable of the equation (dx ).

Comments:

Calculus v. 2015 - 2016

Content: Definite Integrals Duration: March (2 Weeks)

Essential Question: How can the area under a curve be calculated?

Skill:

Define a definite integral (including an explanation of the mathematical formula).

Understand that a definite integral represents a signed area, and therefore can be positive, negative or even zero.

Know and understand the properties of definite integrals.

o ∫a

a

f ( x )dx=0

o ∫a

b

f ( x )dx=−∫b

a

f ( x )dx

o ∫a

b

kf (x )dx=k∫a

b

f ( x )dx , where k is a constant

o ∫a

b

[ f ( x )±g ( x ) ] dx=∫a

b

f ( x ) dx±∫a

b

g ( x )dx

o ∫a

b

f ( x )dx=∫a

c

f ( x )dx+∫c

b

f ( x )dx, where a<c<b.

Approximate the area under a curve using Riemann Sums.o Left (Ln )o Right (Rn )o Average ( An)o Midpoint (M n )

Know and explain the Fundamental Theorem of Calculus. Evaluate a definite integral using the Fundamental Theorem of Calculus.

Assessment:

Given a graph for f ( x ) and the value of the areas on the graph, find the value of a given definite integral using the definite integral definition and properties.

Given a function, an interval and the number or rectangles; use the Left, Right, Average and Midpoint sum to approximate the area under the curve.

o From x = 0 to x = 5, using 4 intervalso From x = 1 to x = 2, using 5 intervals

What is the best approximation to use for finding the area under the curve? How can all of the approximation become more accurate?

Give a brief explanation of the following definition for a definite integral.Let f be a continuous function on [a, b], and let1.a=xo<x1<…, xn−1<xn=b2.∆ xk=xk−xk−1 for k=1,2 ,…n3.∆ xk→0asn→∞4. xk−1≤ck ≤xk for k=1,2 ,…,n

Calculus v. 2015 - 2016

Then ∫a

b

f ( x )dx=¿ limx→∞

∑k=1

n

f (ck)∆ xk ¿

Explain the Fundamental Theorem of Calculus. Use the Fundamental Theorem of Calculus to evaluate the definite integral

of a function.

o ∫1

8 23 3√x

dx

o ∫1

2

( 2x3 + 23 x )dx

o ∫2

6

( 5 x4x5−8 )dx

Resources:Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections 6.4 & 6.5 (pages 401 – 431Calculus: AP Edition. Pearson: Sections 5.2, 5.3 & 5.4 (pages 349 – 393)Graphing Calculator


Vocabulary:

Definite Integral– The cumulative sum of the signed areas bounded by a curve and the x-axis; Riemann Sum– A method for approximating the area under the curve using rectangles

Comments:

Calculus v. 2015 - 2016

Content: Integral Applications Duration: April (2 Weeks)

Essential Question: How can integration be used to interpret the graph of a function?

Skill:

Define the average value of a function and explain using a graphical representation.

Find the average value of a function. Find the area bounded by a curve and the x-axis. Find the area bounded by two curves.

Assessment:

Given a function, find the average value of the function over a specified interval.o y=x−3x2 on the interval [−1,2 ]o y=√x+1 on the interval [3,8 ]o y=3 e2 x on the interval [2,5 ]

Given two functions, find the area bounded between them.o y=x2+2 & y=x−2 on the interval −1≤x ≤3o y=x2+2 x+1 & y=2x−5o y=x−1+3 on the interval −4≤x ≤−1

Resources:Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections 6.5 & 7.1 (pages 427 – 431 & 445 – 455)Calculus: AP Edition. Pearson: Sections 5.5 & 6.2 (pages 393 – 400, 445 – 454)


Vocabulary:

Average Value – The average height of the graph of f ( x ) over the interval [a ,b ]

. As given by the formula: f= 1b−a∫a

b

f ( x )dx

Comments:

Calculus v. 2015 - 2016

Content: Trigonometric Derivative and Integrals

Duration: April (2 Weeks)

Essential Question: How can the derivative and integral of a trigonometric function be calculated?

Skill: Know and understand the rules for differentiating trigonometric functions.

o ddxsin x=cos x

o ddxcos x=−sin x

o ddxtan x=sec2 x

o ddxcsc x=−csc xcot x

o ddxsec x=sec x tan x

o ddxcot x=−csc2 x

Use the trigonometric rules for differentiation in conjunction with prior rules (especially how the chain rule relates).

o ddxsin f ( x )=f ' ( x ) cos f ( x )

o ddxcos f (x )=−f ' ( x )sin f ( x )

o ddxtan f (x )=f ' ( x ) sec2 f ( x )

o ddxcsc f ( x )=−f ' ( x ) csc f ( x )cot f ( x)

o ddxsec f ( x )=f ' (x ) sec f ( x ) tan f ( x )

o ddxcot f ( x )=− f ' (x ) csc2 f ( x )

Know and understand the rules for integrating trigonometric function.o ∫sin x dx=−cos x+co ∫cos xdx=sin x+co ∫ sec2 x dx=tan x+co ∫ csc x cot xdx=−csc x+co ∫ sec x tan x dx=sec x+co ∫ csc2 x dx=−cot x+c

Use integration by substitution to integrate trigonometric functions. Use integration by substitution to develop 4 more trigonometric integration

rules.

Calculus v. 2015 - 2016

o ∫ tan xdx=−ln|cos x|+co ∫ sec x dx= ln|sec x+ tan x|+co ∫cot x dx=ln|sin x|+co ∫ csc x dx=− ln|csc x+cot x|+c

Assessment:

Evaluate the derivative of a trigonometric function.

o f ( x )= cos x3

+ 1sin x

o f ( x )=sec42 x+5o f ( x )=e( sin x ) (cos x )

o f ( x )=ln|csc x2|o f ( x )= e3x

2 tan x

Evaluate: ∫( cos x4 −3sin x )dx Evaluate: ∫ ( sec2 x+ tan x)dx Evaluate: ∫ ( cos2x−csc x ) dx Evaluate: ∫sin 2 x cos xdx Evaluate: ∫ sin e

− x

exdx

Evaluate: ∫ tan x sec3 x dx Evaluate: ∫ (cos x ) [ ln|sin x|]

sin xdx

Resources:

Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections 9.2 & 9.3 (pages 566 – 579)Calculus: AP Edition. Pearson: Sections 3.5, 5.1 & 5.6 (pages 173 – 181, 341, 405)


Vocabulary: No new vocabulary

Comments:

Calculus v. 2015 - 2016

Calculus v. 2015 - 2016

Content: Review of Trigonometric Functions

Duration: May (1 Week)

Essential Question:

How do you find exact values for the six trigonometric functions of special angles?What is the difference between an identity and an equation?How do the graphs of the six trigonometric functions compare?

Skill:

Evaluate the EXACT value of the sine, cosine, tangent, secant, cosecant or cotangent at a given angle in radian measure. (Using the unit circle)

Know and use the trigonometric identities to manipulate and solve an equation.

Graph the six trigonometric functions. Use transformations to graph trigonometric functions.

Assessment:

Completely fill in a unit circle or table of trigonometric values. Use the unit circle or table of trigonometric values to evaluate the sine,

cosine, tangent, secant, cosecant or cotangent of an angle.

o sin( 2 π3 )o tan( 5π6 )o sec (−11π6 )

Use the Reciprocal, Pythagorean, Double and Half angle identities to prove trigonometric equations.

o csc2x−1csc2x

=cos2 x

o sin x1−cos x

=csc x+cot x

o sin2 x=1−cos2 x2

Use the Reciprocal, Pythagorean, Double and Half angle identities to solve trigonometric equations.o 2θ cosθ+θ=0o 2−2cos2 x=sin x+1o sin2θ=2cos2 x

2

o cos2 x−cos 2x=12

Graph the sine, cosine, tangent, secant, cosecant and cotangent functions. Use transformations to graph trigonometric functions.

o Graph y=2sin 3 x+2

o Graph y=−cos(x+ π2 )Resources: Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed.

Prentice Hall; Section 9.1 (pages 557 – 566)

Calculus v. 2015 - 2016

Calculus: AP Edition. Pearson: Section 1.4 (pages 41 – 53)


Vocabulary:

Amplitude– The distance from the mean of a function to the maximum or minimum value; Double Angle Identity– sin 2 x=2 sin x cos x or

cos2 x=cos2 x−sin 2 x ; Half Angle Identity– sin2 x=1−cos (2 x )2

or

cos2 x=1+cos (2 x )2

; Identity– An equation that holds true for all real numbers; Period– The distance it takes for a function to complete one full cycle; Pythagorean Identity– sin2 x+cos2 x=1

Comments:

Calculus v. 2015 - 2016

Content: Derivative Applications of Trigonometric Functions

Duration: May (2 Weeks)

Essential Question:

How can the graph of a trigonometric function be explained/interpreted using techniques of calculus?

Skill:

Write the equation of a tangent line at a given point for a trigonometric function.

Determine where the graph of a trigonometric function has a horizontal tangent line.

Determine where the graph of a trigonometric function is increasing, decreasing and locate the local extrema (First Derivative Test).

Determine where the graph of a trigonometric function is concave up, concave down and locate inflection points.

Use the Second Derivative Test to find the local extrema of a function. Determine the absolute extrema of a trigonometric function on a closed

interval. Solve Related Rate problems involving trigonometric functions. Solve Optimization problems involving trigonometric functions. Evaluate definite integrals involving trigonometric functions. Find the area bounded between two trigonometric functions.

Assessment:

Write the equation of a tangent line to y=4+cot x−2csc x, at x=π2

.

What is the equation of a line tangent to x2cos2 y−sin y=0 at (0 , π). Where does g ( x )=sin2 x, have horizontal tangent lines? (Give all answers

between 0 and 2π ) Where is the function f ( x )=x+sin x, increasing and decreasing. Where does f (x) have local extrema? (Give all answers between 0 and 2π )

Where is h ( x )=e− xsin x, concave down and concave up. Where does h(x ) have inflection points? (Give all answers between 0 and 2π )

Use the second derivative test to locate the local extrema for the function f ( x )=sin x−cos x (Give all answers between 0 and 2π )

A 13 foot ladder is leaning against a house, when its base starts to slide away. By the time the base is 12 feet from the house, the base is moving at a rate of 5 feet per second. At what rate is the angle between the ladder and the ground changing at that moment?

The trough in the figure is to be made to the dimensions shown. Only the angle θ can be varied. What value of θ will maximize the volume of the trough?

Resources:

Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 9.2 (pages 566 – 572)Calculus: AP Edition. Pearson: Sections 3.11, 4.1, 4.2 & 4.4 (pages 232 – 240, 246 – 270 & 281 – 291)

Calculus v. 2015 - 2016



Comments:

Calculus v. 2015 - 2016

Content: Integration Applications of Trigonometric Functions

Duration: May/June (2 Weeks)

Essential Question: How can integration be used to interpret a function and its graph?

Skill:

Evaluate a trigonometric definite integral. Find the area bounded by the graph of a trigonometric function and the x-

axis. Find the area bounded by two trigonometric functions. Find average value of a trigonometric function.

Assessment:

Evaluate ∫π3

2π

cos x dx

Evaluate ∫0

π4sin xcos2 x

dx

Evaluate ∫0

π2

tan x2dx

Find the area bounded by y=x sin x2 and the x-axis between x=0 and x=√ π Find the area bounded by y=sin x cos x and y=0 between x=0 and x=√ π Find the area bounded by y=sin x and y=sin2 x on the interval [0 , π ]

Find the average value of the function f ( x )=sec3 x tan x on the interval [0 , π3 ]

Resources:Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 9.3 (pages 572 – 577)Calculus: AP Edition. Pearson: Sections 5.3 – 5.5 & 6.2 (pages 364 – 400 & 445 – 454)


Calculus v. 2015 - 2016


Comments:

consensus map grade level · web viewgraph a quadratic (standard form or vertex form), rational,...

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