MIDTERM 1 STUDY GUIDE PEYAM RYAN TABRIZIAN Know how to Given a graph or given a formula find values of a function and solve equations such as Find x such that f x 2 1 1 1 Determine if a graph is a graph of a function 1 1 6 Sketch graphs of functions representing real life situations 1 1 14 1 1 18 Find domains and ranges of functions given a graph or given a formula 1 1 30 1 1 32 Quiz 1 1 2 Solve word problems 1 1 57 1 3 55 Quiz 1 5 Know how to draw graphs of linear functions power functions e g x3 or x and exponential functions e g 3x 1 2 4 Use the above functions in word problems 1 2 15 Graph new functions from old ones e g given f graph f x 1 3 5 1 3 6 Explain for example how you can get the graph of f x 2 3 given the graph of f 1 3 2 1 3 3 Quiz 1 3 Compose add multiply and divide functions and find their domains 1 3 33 1 3 39 Compositions represent in real life situations 1 3 55 ex Find domains of functions involving ex e g Find the domain of 1 e x 1 5 15 Determine whether a function is one to one given its graph 1 6 6 or given a formula 1 6 9 1 6 10 Find the inverse of a function given its graph i e reflect about the line y x 1 6 29 1 6 30 Find the inverse of a function given a formula 1 6 25 1 6 26 Know what f 1 4 actually means 1 6 17 Do computations with ln and logs and simplifying expressions involving ln and logs 1 6 33 1 6 36 1 6 39 Find average velocities of a function given a table or given a formula and estimate instantaneous velocities 2 1 6 2 1 7 Find the limit of a function at a point or at infinity or say that it does not exist and vertical horizontal asymptotes of a function given its graph 2 2 7 2 2 9 2 6 3 Sketch the graph of a function with given limits 2 2 15 2 6 7 Given 2 graphs of f and g finding limits of f g f g etc 2 3 2 Evaluating a limit of a function at a point or showing that it does not exist given its equation By substituting into the expression 2 3 3 2 3 5 By noticing for example that it s of the form 01 and hence it s 2 2 25 2 2 28 Date Friday September 17th 2010 1 2 PEYAM RYAN TABRIZIAN By noticing that the left hand limit and the right hand limit are equal or not equal if the limit does not exist 2 3 39 2 3 40 2 3 45 Good for piecewisedefined functions By factoring the numerator denominator and by canceling out 2 3 11 2 3 15 also look at 2 3 61 By multiplying numerator and denominator by a b whenever you see something involving a b 2 3 21 2 3 22 2 3 23 2 3 30 2 3 60 By using the squeeze theorem 2 3 37 2 3 38 Evaluating a limit of a function at infinity or showing that it does not exist and stating its asymptotes given its equation By substituting into the expression 2 6 15 By factoring out the highest power of the numerator and the highest power of the denominator 2 6 16 2 6 19 2 6 21 also 2 6 33 or simply the highest power of the expression 2 6 31 By multiplying numerator and denominator by a b whenever you see something involving a b 2 6 25 2 6 26 2 6 27 By factoring out the highest power of the square root when the preceding method fails 2 6 23 2 6 24 By noticing that the function is bigger than or smaller than a familiar function whose limit you know 2 6 30 Finding limits rigorously using an argument 2 4 19 2 4 22 2 4 29 2 4 30 2 4 31 2 4 32 2 3 36 2 3 37 Find left hand side and right hand side limits rigorously using an epsilon delta argument 2 4 28 Find infinite limits rigorously using an epsilon delta argument 2 4 42 2 4 44 Find limits at infinity rigorously using an epsilon delta argument 2 6 65 2 6 67 This includes infinite limits at infinity Note Be careful Sometimes you may have a problem that does not involve explicitly look at 2 6 13 2 6 14 Given a graph state the numbers at which the function is continuous or not 2 5 3 Given a formula show that a function is continuous at a point 2 5 43 Given a formula find the numbers at which a function is discontinuous 2 5 37 2 5 39 Using the intermediate value theorem to show that an equation has a root or that two functions are equal at a point 2 5 47 2 5 51 Solving word problems using the intermediate value theorem 2 5 65 Calculate the derivative of a given function at a given point using the definition of a derivative 2 7 25 2 7 27 2 7 30 Recognize a certain limit as a derivative of a function 2 7 31 2 7 34 2 7 36 Find the equation of the tangent line to the graph of a given function at a given point 2 7 10 Know what a derivative means in real life in particular find the instantaneous velocity of a particle at a given time 2 7 37 2 7 38 2 7 46 MIDTERM 1 STUDY GUIDE 3 Also know how to define the following terms state the following theorems remember that for functions you ll need to state the domain and the codomain of that function Function Domain of f Range of f Absolute Value Function Increasing Decreasing Vertical line test e 2x more generally ax f g f composed with g Inverse function ln x more generally loga x logr 2 just say it s the number y such that ry 2 limx a f x L the rigorous definition as well as its variants limx a f x L limx a f x and limx f x L limx f x Vertical Horizontal asymptote The Squeeze Theorem f is continuous at a and its variant with left right continuous f is continuous on an interval I The Intermediate Value Theorem The derivative of f at a i e f 0 a both definitions the limit definition and the tangent line definition The tangent line to y f x at P a f a