Calculusv. 2015 - 2016
Calculus
This 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 minutes
Text: Barnett, Raymond, et al. Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Upper Saddle River: Prentice Hall, 2002
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=-23x+3-1
· Graph y=e-x+3
· Graph y=lnx+3-4
· Analyze and graph fx=-3xx+32
· Analyze and graph fx=2x4-5x3-4x2+3x+6
· Solve: x2-5x=6
· Solve: 53x-73+x3x-72=0
· Solve: 2x=5+1x2+x
· Solve: 3+3x+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: fx=bx, where b is any real number such that b>0 aand b≠1; Logarithmic Function – Any function in the form: fx=logbx, where b is any real number such that b>0 aand b≠1; (+ction in the form Any t ical expression in the radacand.ls.he limitrence of cnxponential, XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXPolynomial Function– An function in the form fx=anxn+an-1xn-1…a2x2+ a1x+a0. Where n is an integer and a is a number; Quadratic Function– A function in the form fx=ax2+bx+c or fx=ax-h2+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
Standards: / This content is beyond the scope of the PA Core Mathematical Standards.
Vocabulary: / Continuity– A function is continuous at a point if the following three conditions are met: 1. limx→afx exists; 2. fa exists; 3. limx→afx=fa; 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 fx=2x+3 at x=-4
o tx=3-7x 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 kx=12-x
o Find g'x given gx=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)
Standards: / This content is beyond the scope of the PA Core Mathematical Standards.
Vocabulary: / Average Rate of Change– Slope of a secant line m=fa-fba-b; Derivative– A function that gives the slope of a tangent line at any point in the domain of the function limh→0fx+h-fxh; Difference Quotient– fx+h-fxh; Instantaneous Rate of Change– Slope of a tangent line limh→0fa+h-fah
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 fx=32x+x3
· Find the derivative of fx=lnx+e-x
· Find the derivative of y=log2-x
· Find the derivative of fx=32x+x2
· Find the derivative of y=ln(x2-2x)
· Find the derivative of fx=ex+5lnx2
· Find dydx if x=ey+7
· Given a function fx, find the x values for which the function is non-differentiable and explain why the derivative does not exist at these points.
· 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 fx=3x23-53x
· Write the equation of the tangent line at x=e for the function fx=lnx2
· Given a function, algebraically find where it has a horizontal tangent line. f'x=0
· fx=xx+34
· gx=xlnx
· 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)
Standards: / This content is beyond the scope of the PA Core Mathematical Standards.
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 variablex;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:
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 P200and 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.
Standards: / This content is beyond the scope of the PA Core Mathematical Standards.
Vocabulary: / Average (Cost, Revenue, Profit)– The average cost, revenue or profit per 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:
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 anaccurate graph of a function and its derivative.
Assessment: / · Use the relationship between fx and f'x to answer graphical questions given fx, and f'x.
o Given a graph of fx determine where f'x is negative.
o Given a graph of f'x determine where fx increasing.
o Given a graph of f'x determine where fx has a local minimum.
· Use the relationship between fx, f'x, and f''x to answer graphical questions given fx, f'x, and f''x.
o Given a graph of fx determine where f'x is decreasing.
o Given a graph of f'x determine where f''x positive.
o Given a graph of f''xdetermine where does fx have inflection points.
· Given an equation for fx determine where the function is increasing and decreasing by making a sign chart for f'x.
o fx=3x2-9
o fx=xe-x
· Given an equation for fx use the first derivative test to interpret the sign for f'x and determine where the function has local extrema.
o fx=x23-x53
o fx=lnxx , x>0
· Given an equation for fx determine where the function is concave up, concave down and where inflection points occur by making a sign chart for f''x.
o fx=3x-3 +1
o fx=lnx2-2x+10
· Given an equation for fx use the second derivative test to determine where the function has local extrema.
o fx=x-23-x5
o fx=4+x+9x
· Given a sign chart for f'x and f''x, construct an accurate graph of fx 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 fx.
· Given the graph of fx, 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)
Standards: / This content is beyond the scope of the PA Core Mathematical Standards.
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 fx; 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 fx 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 fx