(Although 1.1-1.6 will not be covered in class, you should also aim to be able to do the objectives listed for those sections.)

• State the definition of function and work with functions given as graphs, equations, tables, and in words.
• Find the domain of a function and sketch its graph.
• Use and construct piecewise functions.
• Construct functions described in words.
• Use graphical symmetries of odd and even functions.

• List and identify all of the basic types of functions.

• Apply vertical and horizontal shifts, stretches, and reflections of functions algebraically and graphically.
• Combine functions by arithmetic operations and by composition.
• Express a given function in the form of a composition by finding separate functions.
  

• Use a graphing calculator effectively by incorporating mathematical knowledge with such use.
• Be aware of pitfalls that can occur when using a graphing calculator without sufficient mathematical understanding.

• Graph exponential functions, find their domain and range and long term behavior.
  
• Use laws of exponents.
• Find exponential functions from given information in graphical and applied settings.

• Determine if a function given algebraically, graphically or in a table is one-to-one and if so, find its inverse.
  
• State and apply the definition of a logarithmic function or equation.
• Apply laws of logarithms to simplify expressions and solve equations.
• Graph and work with inverse trigonometric functions.

• Describe what secant lines and tangent lines are.
• Use the pattern in the slopes of secant lines to guess the slope of a tangent line.
• Compute slopes of secant lines and estimate slopes of tangent lines from tables of data and formulas.
• Find average and instantaneous velocity using slopes of secant and tangent lines.

• State the rough definition and use the notation for the one and two-sided limits of a function f(x) as x approaches a.
• Read limits from graphs and guess at limits from tables of values, possibly constructed using a calculator.
• State the meaning of an infinite limit.
• Determine infinite limits using behavior near vertical asymptotes and find equations of vertical asymptotes using infinite limits.

• Calculate limits using the limit laws and justify the steps.
• Calculate limits of polynomial and rational functions using substitution and algebraic functions using algebra and limit laws or substitution.
• Calculate limits using the Squeeze Theorem.
• Recognize when the limit laws, substitution principle, and Squeeze Theorem cannot be applied because a hypothesis is not satisfied.

• State the precise, rigorous definition of the limit of f(x) as x approaches a.
• Estimate from a graph or a formula a value of delta for a specific epsilon in the definition of limit.
• (For Math Majors: Some advanced math courses assume you are familiar with the rigorous definition.  Try problems from #15-32.)

• Use all three parts of the definition of continuity along with finding limits to explain why a function is continuous or discontinuous at a value of x.
  
• Find locations of continuity or discontinuity from a graph or a formula and explain your answers using theorems on continuity.
  
• Identify types of functions that are continuous on their domains.
• Use the Intermediate Value Theorem to explain why a function must take on certain values.
• Explain why continuity of the function is required in the hypothesis of the Intermediate Value Theorem.

• State the rough definition for the limits of f(x) as x approaches infinity and -infinity.
• Find limits at infinity using graphs and use algebra to find limits at infinity for functions given by formulas.
• State and use the definition of horizontal asymptote.
• Use limits at infinity to find horizontal asymptotes and sketch graphs of functions.

• State and use the precise definition of slope of a tangent line and derivative of a function f at a number a using a limit of a difference quotient.
• Find slopes and equations of tangent lines to simple functions and derivatives of simple functions using the definition.
• Use slopes of secant and tangent lines to find average and instantaneous rates of change in physical situations, including velocities.
• State and use the meaning of the derivative of a function f at a number a (as the slope of the tangent line to f at x=a and as the instantaneous rate of change of f(x) with respect to x at x=a) including in applied problems.
• Estimate values of the derivative at a point using a graph or table.

• Find the derivative of a function f as a function of x from the definition.
• Find or construct graphical information about f '(x) from f(x).
• Explain how f can fail to be differentiable and find points where f fails to be differentiable from the graph of f.
• Use the theorem that if a function f is differentiable, then f is continuous, especially in the contrapositive form, if f is not continuous, then f is not differentiable.
• Know the meanings of higher derivatives of a position function.

• Find the derivative of constant, power, polynomial, and e^x functions.
• Find derivatives using the constant multiple, sum, and difference rules.
• Be able to find the derivative at a point and the equation of a tangent line using the rules above.

• Find derivatives using the product and quotient rules, from a formula, graph, or table.
  
• Be able to find the derivative at a point and the equation of a tangent line using the product and quotient rules.

• Understand the algebraic proofs and the geometric evidence for the differentiation formulas for trigonometric functions.
• Use the trigonometric limits #2 and #3 that arise in the proofs that d/dx(sin x) = cos x and d/dx(cos x) = -sin x to find limits of functions involving trig functions.
• Use the formulas for the derivatives of trigonometric functions to find derivatives and equations of tangent lines.

• Work examples showing why the Chain Rule is needed to differentiate composite functions.
• Recognize composite functions and find their derivatives using the Chain Rule primarily starting from formulas, but also from graphs and tables.
• Find derivatives of general exponential functions.

• Recognize when it is appropriate to use implicit differentiation.
• Use implicit differentiation to find a derivative of one variable (regarded as dependent) with respect to another (regarded as independent), even when we cannot solve for the dependent variable in terms of the independent variable.

• Differentiate functions involving logarithms.
• Use logarithmic differentiation to more easily find derivatives of functions involving products, quotients, and powers.
  

• Give examples of how rates of change are used in other fields such as physics, chemistry, biology, engineering, economics.
  
• Describe meanings and find units of rates of change in many physical situations.
• Calculate rates of change from formulas and use these to answer practical questions.
• Estimate rates of change from tables.

• Recognize why a rate of change in a quantity that is proportional to the quantity is modeled by an exponential function.
• Set up, use, and analyze several models of exponential growth and decay including population growth, radioactive decay, Newton's law of heating and cooling, and compounding interest.

• Apply the strategy for solving related rates problems on p. 243 to solve for the rate of change in one quantity given information about the rate of change in another quantity.

• Find the linear approximation to a function using the interpretation of the tangent line as the best linear approximation.  (In Chapter 11 in Math 175 you will study higher degree polynomial approximations to functions.)
• Use the linear approximation to make predictions and estimations in physical situations.
• Find differentials and understand their geometric interpretation on a graph (used in 5.5).

• Use the definitions of hyperbolic functions to prove identities, find limits, and find derivatives for hyperbolic functions.
• Find and use derivatives of functions involving hyperbolic functions.
• Use the definitions of inverse hyperbolic functions to find derivatives.
  

• State precise definitions of local and absolute maximum and minimum, critical value.
• Identify the various extreme values and their locations on graphs and be able to construct functions when extreme values and related behavior are specified.
• Locate critical values of a function given by a formula.
• State Fermat's Theorem and be able to use its logic correctly to draw conclusions.
• Use the Closed Interval Method to find absolute minimum and maximum values of a continuous function on a closed interval when the function is given by a formula.

• Test the hypotheses for Rolle's Theorem and the Mean Value Theorem and explain how their failure can lead to no such c existing in (a,b).
  
• Use Theorem 5 (proven using the Mean Value Theorem): If f '(x)=0 on (a,b) then f is constant on (a,b).
• Use Corollary 7: If f '(x)=g'(x) on (a,b) then f(x)=g(x)+c.
  
• Be aware that the Mean Value Theorem will be used to prove the Increasing/Decreasing Test in 4.3.

• Use the Increasing/Decreasing Test to find intervals on which a function f given by a formula is increasing or decreasing.
• Use the First Derivative Test to find local maximum and minimum values of f.
• State the definitions of points of inflection, concave up, and concave down.
• Use the Concavity Test to find intervals on which f is concave up or down.
• Use the Second Derivative Test to find local maximum and minimum values of f.
• Specify behavior of a function f when given a graph of f, f ', or f ''.

• Use L'Hospital's Rule to compute limits for indeterminate forms of the types given in the section.
  
• Recognize when L'Hospital's Rule cannot be applied because a limit is not an indeterminate form or another hypothesis fails.

• Effectively use both graphing by hand and graphing by calculator to identify important aspects of the behavior of a function.

• Set up and solve optimization problems.
• Clearly document and explain the steps used in solving an optimization problem.

• State what Newtion's method is used for and why it is important.
• Estimate values and predict outcomes of Newton's method from a graph.
• Perform Newton's method to estimate roots of equations.

• State the definition of antiderivative.
• Find the most general antiderivative of a basic function and a specific antiderivative if given additional information.
• Sketch a direction field and use it to draw an antiderivative of a function.

• Give over- and underestimates for the area under a curve using rectangles, starting from a formula, a graph, or a table.
• Use the behavior of a function to decide whether a left or right hand estimate is an overestimate or an underestimate.
• Write estimates using sigma notation (also see Appendix E).
• Use the area under a velocity curve to find distance travelled.

• State both parts of the Fundamental Theorem of Calculus.
• Use part 1 of the Fundamental Theorem to find derivatives of functions defined by integrals.
• Use part 2 of the Fundamental Theorem to evaluate definite integrals.
• Interpret the Fundamental Theorem as showing that differentiation and integration are opposite processes.

• Find indefinite integrals of basic functions.
• Use the definite integral to find net change.

• Use substitution to evaluate definite integrals, using correct notation.
• Use symmetry to simplify evaluation of definite integrals.
  Mastering the substitution method will pay off dramatically in Math 175!

• Set up a limit of a Riemann sum representing an area between curves and recognize it as a definite integral.
• Draw regions bounded by given equations.
• Find the area of a region bounded by given curves by calculating the intersection points of the two curves and using a definite integral.

• Set up a limit of a Riemann sum representing the volume of a solid with given cross sectional area and recognize it as a definite integral.
• Calculate volumes of solids obtained by rotating a region bounded by given curves around a horizontal or vertical line

• Know why the formula for the volume of a solid using shells works.
• Calculate volumes of solids using a definite integral and cylindrical shells.
• Recognize when it is more useful to use shells than slices to find volumes.

• Know the formula for the average value of a function on an interval and why it holds.
  
• Calculate the average value of a function on an interval using a definite integral.

• If you have any questions regarding the grading or about upcoming homework or anything else you are encouraged to ask before or after class or in office hours.
• Grades in the left margin mean: check-plus=perfect or nearly perfect, check=mostly right, check-minus=mostly wrong, x=nothing or nearly nothing.
  The initial marks and scores were fairly lenient as you get used to what is expected.  As we go along expect them to become more stringent!
  
• On homework please
  
  • start all problems at the left margin, 
  • write your work as clearly as possible including using pencil (or white out errors if using pen),
  • give complete answers including explanations and responses to all parts of the problem,
  • use graph paper for all graphs and label scales, variables, and (in applied problems) meanings and units for those variables on the axes, 
  • round to the number of decimal places listed when requested in the problem (and show this using an approximation sign rather than an equal sign),
  • use correct notation including only connecting items by an equal sign when they are really equal.
• The updated class directory and solutions to selected problems from homework, quizzes, and worksheets are posted in Moodle.
• The Math Center is now open for drop-in tutoring help in Museum 433 M-Th 9am-7pm and F 9am-2pm and in CHE 220 in Idaho Falls M,W 9am-3pm, T 9-9:30am, 10am-4pm, Th 9-9:30am, 10am-3pm, F 10am-1pm.
• Some studying suggestions for the exam: start early, work extra problems from the book including the review at the end of the chapter (it is especially important to work different types of problems mixed together, as opposed to ordered by section, and to work problems without your book or notes available and under some time pressure), go over old homework and quizzes, reread sections and use supplementary materials to make your own outlines of important points, check if you can do the objectives below, come and ask questions in office hours.
• Exam 1 revised: Mean: 73.2  Median: 72.5  High: 95.6
  Your slip shows all the scores I have recorded for you - please check these and bring any errors to my attention.  It estimates your grade in the course by dropping the lowest quiz and the lowest homework scores and weighting your exam score as 76% of your grade.  Later exams will give you the opportunity to change this.
• Remember that in-class work (worksheets and other problems covered in class and turned in for points based on effort) counts as 8% of your grade.
• For the Mastery Quiz you should be able to work straightforward differentiaion problems involving the constant mutliple, sum, difference, product, quotient, and chain rules applied to all basic types of functions (powers, polynomials, exponentials, logarithms, trig functions, inverse trig functions, hyperbolic functions, and inverse hyperbolic functions).  If you do not get at least 7 of the 8 problems perfect then you will need to take a new version of the quiz outside of class time until you do so.  A good way to study for the quiz is to work as many of the 50 differentiation problems in the Chapter 3 review as possible.
• Make sure to always include units in answers to applied problems - consider the NASA Mars Lander example!
• Midterm grades were submitted for those estimated to be earning below a C- in the course - please check your MyISU.  If you received one of these you are especially encouraged to come ask me questions individually to help improve your grade as I believe all of you are capable of succeeding in the course given sufficient and consistent hard work.
• You may submit a problem you make up (not just taken from the book), that is at a medium level of difficulty and testing one of the objectives listed for one of the sections on the exam.  You will earn up to 2 points toward your in-class score and your problem may be chosen to be on the exam depending on the appropriateness and creativity of the question.