STUDY GUIDE FOR THE FINAL
PEYAM RYAN TABRIZIAN
Note: Please also look at the ’Definitions-Theorems-Proofs’-handout for a list of definitions and proofs you should know for the final!
• Know how to:
1. C HAPTER 1: F UNCTIONS AND M ODELS
• Find domains and ranges of functions, given a graph or given a formula (1.1.30,
1.1.32)
√
• Know how to draw graphs of linear functions, power functions (e.g. x3 or x),
and exponential functions (e.g. 3x ) (1.2.4)
• Graph new functions from old ones, e.g. given f , graph f (−x) (1.3.5, 1.3.6)
• 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 a graph or given a formula (1.6.25, 1.6.26,
1.6.29, 1.6.30)
• Know what f −1 (4) actually means (1.6.17)
• Do computations with ln and logs (1.6.33, 1.6.36, 1.6.39)
Date: Thursday, December 2nd, 2010.
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PEYAM RYAN TABRIZIAN
2. C HAPTER 2: L IMITS AND D ERIVATIVES
• Sketch the graph of a function with given limits (2.2.15, 2.6.7)
• Evaluate a limit of a function at a point (or showing that it does not exist),
given a formula:
– 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)
– 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)
– By using L’Hopital’s rule (see below)
• Evaluate 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)
– By using L’Hopital’s rule (see below)
• 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-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!)
• 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)
• Use 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)
• Solve word problems using the intermediate value theorem (2.5.65)
• Recognize a certain limit as a derivative of a function (2.7.31, 2.7.34, 2.7.36)
• 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)
• Given the graph of a function, sketch the graph of its derivative (2.8.5)
• Given 3 graphs, determine which one is the graph of f , f 0 , f 00 (2.8.41)
• Find the derivative of a function using the definition of the derivative (2.7.25,
2.7.27, 2.7.30, 2.8.21, 2.8.28)
STUDY GUIDE FOR THE FINAL
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3. C HAPTER 3: D IFFERENTIATION RULES
• Know how to find the derivative of a function
(1) Using the power
rule ((xn )0 = n(xn−1 ), valid for all nonzero numbers n,
√
1
even 2 , or 2) (3.1.6, 3.1.8)
(2) Using the sum rule ((f +g)0 = f 0 +g 0 ) and the constant multiple rule ((cf )0 =
c(f 0 )) (3.1.23)
(3) Using (ex )0 = ex , as well as (ax )0 = ln(a) · ax and (ln(x))0 = x1 (3.1.17,
3.1.32, 3.6.3, 3.6.16, 3.6.18)
(4) Using the product rule ((f g)0 = (f 0 )g + f (g 0 )) and the quotient rule (( fg )0 =
(f 0 )g−f (g 0 )
)
g2
•
•
•
•
•
•
•
•
•
•
•
•
(3.2.15, 3.2.18, 3.2.24, 3.2.26)
(5) Using derivatives of trigonometric functions ((cos)0 = − sin, (sin)0 = cos,
(tan)0 = sec2 ) (3.3.10, 3.3.12, 3.3.24)
(6) Using the chain rule ((f ◦ g)0 (x) = g 0 (x) · f 0 (g(x))) (3.4.5, 3.4.13, 3.4.29,
3.4.42, 3.4.46, 3.4.50, 3.4.71)
(7) Using implicit differentiation (3.5.11, 3.5.18, 3.5.27, 3.5.36, 3.5.54)
(8) Using logarithmic differentiation (3.6.30, 3.6.41, 3.6.42, 3.6.50)
Note: Be sure to know how to combine those methods, and THINK about your
problem before you tackle it!
Find the equation of the tangent line to a graph at a point (3.1.35, 3.2.32, 3.3.24,
3.4.54, 3.5.28, 3.6.33)
Find the equation of the normal line to a graph at a point (3.1.35)
Find numbers where a tangent line to a graph is horizontal, or parallel to a given
line (3.1.51, 3.1.55)
Find nth derivatives of functions (3.1.62)
Solve word problems using derivatives (3.3.5, 3.3.37, 3.4.82, 3.7.10, 3.7.18), basically, derivatives represent rates of change
= 1 (3.3.39, 3.3.46, 3.3.51)
Solve problems using limx→0 sin(x)
x
Solve the differential equation y 0 = ky (i.e. y = Cekx ), and use that formula in
real-life situations (3.8.3, 3.8.5)
Using y 0 = ky and other information, find, for example C, or k, or y(something)
or the half-life of an element (3.8.9, 3.8.10)
Solve problems using Newton’s law of cooling (3.8.13, 3.8.15)
Solve related rates problems (3.9.13, 3.9.17, 3.9.18, 3.9.24, 3.9.36, 3.9.44)
Find the linear approximation of a function f at a given point
√ a (3.10.1, 3.10.3)
Use a linear approximation to estimate a given number, e.g. 98 (3.10.25, 3.10.27,
3.10.38)
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PEYAM RYAN TABRIZIAN
4. C HAPTER 4: A PPLICATIONS OF D IFFERENTIATION
• Sketch the graph of a function with given properties having to do with local/absolute
max/min (4.1.7, 4.1.10, 4.1.11, 4.1.14)
• Find the critical numbers of a given function (4.1.33, 4.1.35, 4.1.42, 4.1.44)
• Find the absolute max/min of a given function on a given closed interval (4.1.47,
4.1.51, 4.1.55, 4.1.61, 4.1.62, 4.1.63)
• Use the Mean Value Theorem to:
– Show that an equation has at most one/two roots (4.2.19, 4.2.20, 4.2.22)
– Show that an equation has exactly one root, using in addition the Intermediate
Value Theorem (4.2.17, 4.2.18)
– Estimate the value of a function (look at 4.2.23, 4.2.24, 4.2.25, 4.2.26)
– Solve other problems using the mean value theorem (4.2.28, 4.2.29, 4.2.35,
4.2.36)
• Show that two functions f and g are equal by differentiating them and plugging in
one value for x (4.2.32, 4.2.33)
• Show that f < g by considering h = f − g, and differentiating h (4.2.27, 4.3.74)
• Find intervals on which f is increasing/decreasing, finding the local max/min of
f , as well as intervals of concavity and inflection points, given a graph or given a
formula (4.3.8, 4.3.11, 4.3.13)
• Find a local max/min using the First and/or Second derivative tests (4.3.19)
• Sketch the graph of a function with given properties having to do with first/second
derivatives (4.3.24, 4.3.26)
• Calculate limits using l’Hopital’s rule or another method!!! (any problem between 4.4.5 and 4.4.64 would do)
• Sketch the graph of a given function f , using the DISAIC-method, making sure
to label your graph (4.5.5, 4.5.15, 4.5.21, 4.5.38, 4.5.52, 4.5.56):
– Domain
– Intercepts (x- and y- intercepts)
– Symmetry (even, odd, periodic)
– Asymptotes (horizontal, vertical, slant)
– Increasing/Decreasing/Local Max/Min (Calculate f 0 )
– Concave Up/Down/Inflection Points (calculate f 00 )
• Show that a given line is the slant asymptote to f at ∞ or −∞ (4.5.67, 4.5.68)
• Find the slant asymptotes to f at ∞ or −∞, or show f doesn’t have any slant
asymptotes (4.5.64, 4.5.65)
• Solve optimization problems (4.7.3, 4.7.16, 4.7.17, 4.7.19, 4.7.22, 4.7.25, 4.7.37,
4.7.46, 4.7.53, 4.7.67)
• Determine graphically what happens if Newton’s method is used on a given function (4.8.4)
• Apply Newton’s method once or twice on a given function f with a given initial
value of x0 (4.8.5)
• Apply Newton’s method to approximate a given number (4.8.11, 4.8.12)
• Apply Newton’s method to approximate a root to an equation (4.8.21)
• Find the most general antiderivative of a function (4.9.7, 4.9.13, 4.9.20)
• Find an antiderivative that satisfies a given condition (4.9.29, 4.9.32, 4.9.39)
• Sketch the graph of f , given the graph of f 0 (4.9.51, 4.9.53)
• Find the position of a particle, given its velocity or acceleration (4.9.57, 4.9.61)
STUDY GUIDE FOR THE FINAL
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5. C HAPTER 5: I NTEGRATION
• Estimate the integral of a function, using right endpoints, left endpoints, or midpoints, given n (5.1.3, 5.1.5, 5.2.1, 5.2.2, 5.2.3)
• Express the area under the graph of a given function as a limit (5.1.17, 5.1.18,
5.1.19)
• Determine a region whose area is equal to a given limit (5.1.20, 5.1.21)
• Express a limit as a definite integral on a given interval (5.2.17, 5.2.19, 5.2.20)
• Evaluate integrals directly, using the definition (5.2.22, 5.2.25)
• Evaluate integrals by interpreting them in terms of areas (5.2.34, 5.2.35, 5.2.37,
5.2.39)
• Write a sum of integrals as a single integral (5.2.47)
• Find an integral, given other integrals (look at 5.2.48, 5.2.49)
• Use the comparison property of integrals to verify an inequality (5.2.53, 5.2.54)
• Use the comparison property of integrals to estimate the value of an integral
(5.2.59)
• Show that a function is not integrable (5.2.67, 5.2.68)
• Express a limit as a definite integral and evaluate that integral (5.2.69, 5.2.70,
5.3.65, 5.3.66)
• Use the FTC Part I to find the derivative of a function (5.3.9, 5.3.13, 5.3.16,
5.3.18, 5.3.53, 5.3.56, 5.4.68)
• Use the FTC Part II to evaluate integrals (any problem between 5.3.19 and
5.3.42, as well as between 5.4.21 and 5.4.44, would do)
• Solve slightly more complicated problems with integrals (look at 5.3.73, 5.3.74)
• Find indefinite integrals (5.4.9, 5.4.16, 5.4.17, 5.4.18)
• Interpret what definite integrals represent in real life (5.4.51, 5.4.52)
• Evaluate integrals by using the substitution rule (any problem between 5.5.7 and
5.5.46, as well as between 5.5.51 and 5.5.70 would do)
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PEYAM RYAN TABRIZIAN
6. C HAPTER 6: A PPLICATIONS OF I NTEGRATION
• Use integrals to find areas between curves (any problem between 6.1.5 and 6.1.28
would do)
• Evaluate an integral and interpret it as the area of a region (6.1.31)
• Find volumes of solids obtained by rotating a given region about a specified line,
using the disk, washer, or shell methods (any problem between 6.2.1 and 6.2.36
and between 6.3.3 and 6.3.20 would do)
• Find volumes of ’famous’ solids, such as a cone (6.2.49, 6.2.50, 6.2.51, 6.2.63,
6.3.43, 6.3.44, 6.3.45)
• Find the volume common to two solids (6.2.66, 6.2.67)
• Describe a solid whole volume is represented by a given integral (6.3.29, 6.3.30)
• Find a volume using the disk, washer, or shell method (6.3.37, 6.3.38, 6.3.39,
6.3.40, 6.3.41, 6.3.42)
• Find the average value of a function on a given interval (6.5.1, 6.5.7)
• Solve problems using the MVT for integrals (6.5.13, 6.5.14)