MIDTERM 1 STUDY GUIDEPEYAM RYAN TABRIZIANKnow how to:• Given a graph or given a formula, find values of a function, and solve equationssuch 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. x3or√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 graphof 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)• Find domains of functions involving ex, e.g. Find the domain ofex1+ex(1.5.15)• Determine whether a function is one-to-one, given its graph (1.6.6), or given aformula (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 andlogs (1.6.33, 1.6.36, 1.6.39)• Find average velocities of a function, given a table or given a formula, and estimateinstantaneous 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) andvertical/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 form10+(and hence it’s +∞) (2.2.25,2.2.28)Date: Friday, September 17th, 2010.12 PEYAM RYAN TABRIZIAN– By noticing that the left-hand limit and the right-hand limit are equal, or notequal, if the limit does not exist (2.3.39, 2.3.40, 2.3.45) Good for piecewise-defined 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 seesomething 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 powerof the denominator (2.6.16, 2.6.19, 2.6.21, also 2.6.33), or simply the highestpower of the expression (2.6.31)– By multiplying numerator and denominator by√a − b, whenever you seesomething involving√a + b (2.6.25, 2.6.26, 2.6.27)– By factoring out the highest power of the square root, when the precedingmethod fails (2.6.23, 2.6.24)– By noticing that the function is bigger than or smaller than a familiar functionwhose limit you know (2.6.30)• Finding limits rigorously, using an  − δ-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-deltaargument (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 mayhave 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 thattwo 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 ofa 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 givenpoint (2.7.10)• Know what a derivative means in real life, in particular find the instantaneousvelocity of a particle at a given time (2.7.37, 2.7.38, 2.7.46)MIDTERM 1 STUDY GUIDE 3Also, know how to define the following terms / state the following theorems (rememberthat 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→af(x) = L (the rigorous definition), as well as its variants limx→a+f(x) =L, limx→af(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. f0(a) (both definitions: the limit-definition, and thetangent-line definition)• The tangent line to y = f(x) at P = (a,