MATH 1040 Objectives List

Transcription

1 MATH 1040 Objectives List Textbook: Calculus, Early Transcendentals, 7th edition, James Stewart Students should expect test questions that require synthesis of these objectives. Unit 1 WebAssign problems QP 1, 2, 3, 4 (1.1) Work with the 4 presentations of a function QP1.005, QP2.004, QP2.005, QP2.006, QP2.008, QP2.009, QP2.012, QP3.011, , , , , Determine whether an presentation is a Function Including Vertical line test Evaluate a function at a number, with a variable, with an expression Determine domain and range from a graph Determine domain from algebraic expression Understand the basic absolute value function piecewise definition (will come back to in QP 21) Graph a piecewise function with linear pieces (will come back to in QP 21) Know and use the Even and Odd function definitions (will come back to in App D and QP 24) QP1.001, QP1.004, 1.1JIT.003 QP1.008, QP1.012, QP3.003, QP3.005, QP3.006, QP3.008 QP4.008, QP4.009, QP4.010, QP4.012, , , QP4.002, QP4.003, QP4.005, QP4.006, QP4.007, , , , , , QP 22, 23 (1.3) Add, subtract, multiply, divide functions State domain of each Compose two or three functions State domain of composition function Evaluate function composition QP22.001, QP22.002, QP22.003, QP22.004, QP22.006, QP22.008, QP22.010, QP23.005, QP23.006, QP23.007, QP23.008, QP23.009, , , , , QP23.001, QP23.002, QP23.003, , Decompose two or three functions QP23.011, QP23.012, , , , QP 24 (1.3) Graph the basic unshifted graphs listed here, state domain, range, basic information used in properties: linear, absolute value, x^2, x^3, square root, cube root, problems below 1/x, 1/x^2 (will come back to many of these in detail as we go) Shift, reflect, and scale the basic graphs listed above. (less concentration on scaling than the others) QP24.001, QP24.002, QP24.003, QP24.004, QP24.05, QP24.006, QP24.007, QP24.008, QP24.009, , , , , Determine if a function is even or odd graphically and algebraically QP24.010, QP24.011, QP , QP 5, 6, App B (1.2) Know and use the Distance formula A.B.004, A.B.006, A.B.011, A.B.012 Know and use formula for slope and two forms of a line. Work problems related to these. QP6.003, QP6.005, QP6.006, QP6.008, QP6.009, QP6.014, QP6.015, A.B.010, A.B.021, A.B.022, A.B.026, A.B.027, A.B.029, , , , , ,

2 Know and use properties related to parallel and perpendicular lines. Work problems related to these. Graph lines of form Work word problems related to lines including interpretations Solve linear equations Solve an equation for a certain variable QP6.007, QP6.010, QP6.011, QP6.012, A.B.032, A.B.033, A.B.035,A.B.058 QP6.013, A.B.018, A.B.020, A.B.042 A.B.062 QP5.005, QP5.007, QP5.008, QP5.010, QP5.011, QP5.012 QP5.001, QP5.002, QP5.003, 1.2JIT.001, 1.2JIT.002 QP 7, 8, 9, App C (1.2) Graph parabolas: basic, reflected, shifted, scaled, sideways Write equations given graph Complete square to find parabola vertex and vertex form Solve quadratic by factoring and quadratic formula know and use terminology: roots, intercepts Know and use the results of the discriminant realize imaginary numbers and imaginary solutions exist but we do not concentrate on them in Calc I Write equations for and graph circles and semi-circles Shade region between functions: lines, parabolas, half circles QP7.001, QP7.002, QP7.004, QP7.006, QP7.007, QP7.008, QP7.009, QP7.010, QP7.011, QP7.012 QP8.004, QP8.005 QP8.001, QP8.002, QP9.009, QP9.005, QP9.006, QP8.006, QP8.007, QP8.008, QP8.009, QP8.010, QP8.011, QP10.005, QP A.C.002, A.C.003 A.C.033, A.C.034 Add, subtract, multiply polynomials QP9.009, QP9.011, QP9.012, QP9.013, QP9.014, QP9.015 Graph for positive integer n State domain, range, end behavior QP 10, 11, 12 (1.2) Factor out common terms QP10.003, QP10.004, Factor expressions resulting from product rule Factor with difference of squares, difference of cubes, sum of cubes Factor quadratic like equations with a different powers like Factor by grouping Answer "Is this x-value a root of the polynomial?" Write polynomial whose roots are... Simplify the polynomial Find roots by factoring Factor completely with the assistance of polynomial long division. (Given root or make a guess) Use of synthetic division is acceptable Note: OK to use rational roots test to help decide good guess but we will not ask students to list all possible rational roots Perform polynomial Long Division with more than just where a is a root QP10.009, QP QP10.006, QP10.012, QP8.012 QP QP11.008, QP12.012, QP QP11.002, QP11.004, QP QP12.001, QP12.003, QP QP12.002, QP12.006, QP QP 13, 14 (1.2) Determine domain, roots, and y-intercept for rational function QP13.002, QP13.003, QP Simplify fractional expressions involving rational functions Simplify fractional expressions that have more than 1 variable Note: be sure to do some problems with simplifying by dividing by monomial in denominator Graph and for positive integer n QP13.007, QP13.008, QP13.009, QP13.010, QP13.011, QP13.012, QP13.013, QP13.014, QP QP14.001, QP14.003, QP14.005, QP QP13.001, QP14.011, QP State domain, range, asymptotes, end behavior Determine domain of more complicated functions with including in denominator or QP14.008, QP14.009, QP14.010

3 1.2 Once related QP sections are done, classify type of function given algebraic presentation Note: students will have to read the text in 1.2 about exponentials and logs. We will cover both in much more detail later , , 1.2.JIT.008, 1.2.JIT.010 QP 21 (1.3) Sketch Absolute Value Functions. State domain, range, etc., QP21.002, QP be able to write as a piecewise function. Also sketch shifted absolute value, domain, range, be able to write as piecewise function Graph piecewise functions with linear and quadratic pieces. QP21.005, QP21.006, QP21.007, Domain, Range, evaluation. QP21.008, QP Write a piecewise function for graph with linear and quadratic pieces Learn about rational function that reduces to linear with hole. (we will not test this now.) QP App D, QP 18, 19 Convert back and forth between radians and degrees locate angles on graph Evaluate trig functions at axes points and common angles Should be able to find all 6 trig functions values at these points Evaluate trig at negative, positives, less than 2pi, greater than 2pi A.D.002, A.D.003, A.D.008, A.D.009, A.D.012, A.D.017, A.D.019, A.D.020, A.D.021 QP18.001, QP18.002, QP18.003, QP18.004, QP18.005, QP18.006, QP18.008, QP18.009, QP18.011, A.D.023, A.D.024, A.D.025, A.D.028 QP18.007, QP18.008, QP18.010, QP18.011, QP18.012, QP20.006, QP Know, sketch, and use basic graph of,, and Basic information will be used in many places Know and use the other three trig graphs Basic information will be used in many places Determine if a trig function is Even or Odd Note: there is one problem in WebAssign concerning Law of Cosines A.D.084 for students to research on their own QP 24 w trig (problems in 18, 19) Transform sine and cosine: shift left/right, vertical shift, change period, change amplitude Answer questions about the above functions and graph by hand Transform tangent, cotangent, cosecant, secant: shift left/right, vertical shift, change period, change amplitude (when applicable) Concentration is on sine and cosine , , QP19.004, QP19.005, QP19.007, A.D.077, A.D.082, A.D.081 QP19.008, QP19.009, A.D.078, A.D.080

4 Unit 2 WebAssign problems Stewart book problems App D, QP 20 Given one trig function and a quadrant, find the other five QP20.001, QP20.002, A.D.03, A.D.032, A.D.033 Know, be able to prove, and be able to use the 3 Pythagorean trig Basic information will be used in identities. various places Use the Addition and Subtraction Formulas (will be given if needed on a test) Know and use the Double Angle Formulas and the Half Angle QP Formulas (will NOT be given if needed on test) Prove some trig statements try not to be too hard, but what QP20.003, QP instructors think is hard and what students think is hard... Solve equations involving trig try not to be too hard, but what A.D.065, A.D.066, A.D.067, instructors think is hard and what students think is hard... A.D.068, A.D , QP 16 Graph exponentials and natural exponential functions. Know and use properties including stating domain and range and performing function evaluation. QP16.003, QP16.004, QP16.006, QP16.007, QP16.008, , , , 1.5.JIT.005, , , , Shift exponential and natural exponential functions and graphs , #12, 1.5 #15 Work with exponentials and natural exponential algebraically QP16.001, QP16.002, QP16.005, , Solve exponential equations with the one to one property QP16.009, QP16.010, QP16.011, QP Simplify an expression using Laws of Exponents , #2, 1.5 #4 1.5 #7 1.6, QP 25 Datermine if a function is one to one (graphically with help of horizontal line test) QP25.001, QP25.003, QP25.004, QP25.011, QP25.012, , , , , , Know and use definitions and properties of inverse functions Know properties of and determine Domain and Range of inverse functions Perform evaluations with inverse functions: from table, QP25.005, , algebraically Find inverse graphically and algebraically QP25.006, QP25.007, QP25.008, QP25.009, QP25.010, QP25.014, QP25.015, , , , State a restricted domain where function is 1-1 QP , QP 17 Understand and use logarithms as inverses of exponentials 1.6.JIT.006, , QP17.001, QP17.002, QP Graph logarithmic and natural logarithmic functions. Know and use , QP properties including stating domain and range and performing function evaluation. Shift logarithmix and natural logarithmic functions and graphs QP17.004, QP Work with logs and natural log properties algebraically 1.6.JIT.007, , , , , QP17.008, QP17.009, QP17.010, QP Solve equations involving logs QP17.012, QP17.013, QP17.014, QP Evaluate Inverse trig expressions , , , , Sketch a trigonometric function and its inverse Find the average rate of change of a function over a given interval, 1, 3 and use it to estimate the slope of the tangent line. Using technology, calculate the average velocity of shorter and 5 shorter time intervals in order to estimate the instantaneous 2.2 Explain what a limit means in ordinary language. 1

5 Given a graph of a function, find limits, or explain why a limit does 7, 9 5 not exist. Using technology, calculate the value of a function closer and 21 closer to a given input value in order to estimate the limit. Given select function values and limit statements about a function, 17 sketch a graph with the given properties. Recognize and evaluate an infinite limit. 29, 31, 33, Sketch a piecewise function and determine limits. 11 QP 15 Rationalizing expressions with conjugatation QP15.003, QP15.005, QP15.007, QP15.008, QP15.009, QP Know and use properties of fractional and negative exponents. Simplify expressions with fractional and negative exponents. Instructor supplement 2.3 Find the limit of a function using the algebraic techniques and 1, 11, 13, 15, 17, 21, 23, 29, 41 3 Limit Laws. Find the limit of a function using the Squeeze Theorem. 37 Apply theorems of limits to find unknown limits Given ε and the limit value, use a graph to find a specific δ that 1, 3 satisfies the ε-δ definition of a limit. Apply the ε-δ definition of a limit to a real world problem. 11 Prove a limit using the ε-δ definition. 15, 19, Find a value for δ in an infinite limit problem Given a graph, determine the points at which the function has discontinuities. 4 Given a function, determine the points at which the function has discontinuities. Determine if a function is continuous at a point by definition Find the points (intervals) at which a function is continuous. Justify with continuity theorems. 4, Use the Intermediate Value Theorem to show that an equation has 51 53, 55 at least one solution or a given number of solutions. Determine the value of a constant so that a function is continuous 23, 45 at a point or on an interval. Verify or classify discontinuities. 43 Sketch a function that satisfies given criteria. 5, 7 Relate continuity to a real world problem. 9 Apply theorems of limits and continuity to find unknown function 11 values.

6 Unit 3 WebAssign problems Stewart book problems 2.6 Find a limit at infinity for a given function. 15, 17, 19, 23, 31, 33, 35 Find limits at infinity using a given graph. 3 Find all vertical and horizontal asymptotes of a function. 43 Given select function values and limit statements about a function, 7 9 sketch a graph with the given properties. Apply the Squeeze Theorem to a limit at infinity problem. 61 QP 3b Simplifying the two versions of difference quotient QP , QP Find the value of the derivative of a function at a given value of x 3 1 using one of the limit definitions. Find the equation for a tangent line to a curve at a given point. 5 Use the derivative definition to find instantaneous velocity. 13 Given a graph of a function, approximate the derivative at various 17 points. Determine slopes and function values using a give tangent line. 19 Sketch a function that satisfies given conditions on f and f. 21 Recognize a given limit as a derivative. 33, 35, 37 Determine average rates of change and instantaneous rates of 45, 51 change for a given real-world function. 2.8 Find the derivative function using the limit definition. 21, 23, 27 Given a graph of a function, sketch a graph of its derivative. 5, 9, 11, 13, 15 Interpret the derivative if the function models real-world phenomena. Match the graph of a function with the graph of its derivative. 3 Given a graph, find the values where a function is not 37, 39 differentiable. Given a graph, analyze the slope of the curve at given points. 1 Given three graphs, determine which one is position, which one is velocity, and which one is acceleration Find the derivative of a function using the differentiation rules. 3, 5, 7, 9, 11, 13, 15, 19, 23, 29, 17, 21, Find the second derivative of a function. 43 Find the equation for a tangent or normal line to a curve Find equations for the tangents to a curve having specified slope , 55 Find velocity and acceleration functions for a given position 47 function. Given certain conditions, determine parameters for a quadratic function Find the derivative of a function using the differentiation rules. 9, 11, 13, 15, 21 3, 5, 7, 17, 19, 23, 51 Find the tangent or normal lines to a curve at a given point. 31, 33 Find higher-order derivatives of a function. Given values for functions and their derivatives at a point, find the 43, 45 value of a derivative at that given point for a new function. Given values for functions and their derivatives at a point, find the 47 tangent line at that given point for a new function. 3.3 Evaluate a limit involving (sin x)/x or (cos x 1)/x. 39, Differentiate a function involving trigonometric functions odd, 15, 29, Find the tangents to a curve at a given point. Graph the curve and the tangent line on the same set of axes. Determine if and where a graph has a tangent line of given slope. 33 Use derivatives to answer questions involving the position, velocity, 35 and acceleration of an object 3.4 Use the Chain Rule to differentiate a composite function. 1, 5, 7, 13, 15, 17, 21,23, odd, 47, 61, 63, 71 3, 9,11, 19, 25

7 Find the tangents to a curve at a given point. 51, 53 Determine where a graph has a given slope. 59 Use a graph to evaluate the derivative of a composite function a Use implicit differentiation to find dy/dx 1, 5-21 odd 57 Use implicit differentiation to find the second derivative. 35 Find the lines that are tangent and/or normal to a curve at a 25, 27 given point. Find where a curve has a given slope b Find derivatives involving inverse trig functions odd Unit 4 WebAssign problems Stewart book problems 3.6 Find the tangent to a logarithmic function at a given point. 33 Find the derivative of a logarithmic function or functions involving 3, 7-23 odd, 27, 29 5 logarithms. Use logarithmic differentiation to find the derivative. 39, 43, 45