The Final Exam will be held in class on Monday December 13. It will be comprehensive. A more detailed description of each section, with points to pay attention to, is provided below. And a special key to the topics is:
KEY
MEANING
G
"general" material that may not be tested on explicitly but which you should know in order to be able to handle other questions
T
"theoretical" material, i.e. material that is conceptual in nature
C
"computational" material, i.e. techniques you should be able to execute
1.1 Four Ways to Represent a Function (G)
- domain and range, piecewise defined functions, even or odd, etc
1.2 Mathematical Models (G)
- know what is meant by a polynomial, and also know the standard graphs for y = x2, y = x3, y = x4, etc.
- also know the standard graphs for y = x1/2, y = x1/3, y = 1/x, y = 1/x2, etc.
- know the standard graphs for y = sinx, y = cosx, y = tanx
1.4 New Functions from Old Functions (G)
- know how to translate, stretch, and reflect the graphs of functions to get the graphs of other functions
- know what is meant by the composition of functions
1.5 Exponential Functions (G)
- know the standard graphs for y = ax and y = a-x and their domains, ranges, and special values
- make very sure you now your rules for exponents
- know how the graphs for y = 2x, y = ex, y = 3x, etc., compare to one another
- know how the graphs for y = (1/2)x = 2-x, y = (1/e)x = e-x, y = (1/3)x = 3-x etc., compare to one another
1.6 Inverse Functions and Logarithms (T, G)
- know what is meant by an inverse function, i.e. f and f -1 cancel each othe out when composed, and how to find it
- know how the domain and range of f -1 are related to those of f
- know how to get the graph of f -1 from that of f
- recall the definition of logax as the inverse of ax and how to use this in solving equations involving exponentials and logs
- know the standard graph for y = logax and its domain, range, and special values
2.1 The Tangent and Velocity Problems (C)
- know how to compute the secant from P to Q to a graph and how the slope of this secant relates to the slope of the tangent at P
- know how to compute the average velocity from t to t + h and how it relates to the instantaneous velocity at t
2.2 Limit of a Function (T)
- know the definition of limit in terms of f(x) "approaching" a number L as x "approaches" the number a
- be able to use this definition to pick off limits from a function's graph
- know what is meant by one-sided limits
- know what are meant by infinite limits and how they relate to vertical asymptotes
2.3 Calculating Limits Using the Limit Laws (C)
- know the limit laws and how to apply them
- know your list of special known limits (see page 104, in particular)
- be sure you know how to compute all sorts of limits by first simplifying the expression before applying the rules
- know how to find one-sided limits and how they are related to true limits
2.5 Continuity (T)
- know what is meant by continuity of a function and the ways in which it can break down
- know what continuity means graphically and what the three types of discontinuities are
- know the rules for forming new continuous functions from known ones, as well as a list of known continuous functions
- know the continuity rule for composite functions
- know the Intermediate Value Theorem and how to apply it
2.6 Infinite Limits; Limits at Infinity and Horizontal Asymptotes (T, C)
- know how infinite limits relate graphically to vertical asymptotes
- know how to find horizontal asymptotes by computing limits at infinity (there are techniques here!)
- know what infinite limits at infinity are, how to justify what they are, and how they relate to the graph of a function
2.8 Derivatives (T)
- know the definition of a derivative as a limit quotient and how to compute it in specific examples
- know the interpretation of the derivative as a rate of change and how to use it in word problems
2.9 The Derivative as a Function (T)
- know how the graph of f' is related to that of f
- definition of the concept of "differentiable" and how it relates to the graph of a function (i.e. how can a function fail to be differentiable?)
3.1 Derivatives of Polynomials and Exponential Functions (G, C)
- know the basic rules concerning the derivatives of constant multiples of functions and sums and differences of functions, and how to use them
- know how to differentiate polynomials, and why polynomials are differentiable everywhere
- know the rules for derivatives of exponentials
3.2 The Product and Quotient Rules (G, C)
- know the product and quotient rules for differentiation and how to use them
3.4 Derivatives of Trigonometric Functions (G, C)
- know the derivatives of sin x, cos x, tan x, sec x, csc x, cot x
- know the special limit that sin x /x approaches 1 as x approaches 0 and how to use it in computing related limits
3.5 The Chain Rule (G, C)
- know how to use the chain rule in differentiating composite functions (including triple compositions)
- know the derivative of f(x) = ax
3.6 Implicit Differentiation (C)
- know how to find the derivative of a function that is only defined implicitly, and use it in solving slope problems (i.e. find the equations of the tangent line, etc.)
- know the derivatives of the inverse trig functions
3.7 Higher Derivatives (T)
- be familiar with computing higher derivatives through repeated differentiation
- understand the interpretation of the second derivative as an acceleration and know how to solve problems using it
- be able to distinguish between the graphs of f, f', f'', etc.
3.8 Derivatives of Logarithmic Functions (G, C)
- know the formulas for the derivatives of lnx and logax and ln|x|
- know what is meant by the method of logarithmic differentiation and when it is a useful method
- be able to apply logarithmic differentiation to finding derivatives of product and quotients of powers of polynomials
- be able to apply logarithmic differentiation to finding derivatives of expressions of the form f(x)g(x)
3.10 Related Rates (C)
- know how to apply the steps involved in related rates problems
- determine the important variables (usually those for which you are given or must find a rate of change)
- write out the data given to you concerning the variables and what you want to find
- draw a picture and write out an equation connecting the variables (pythag theorem, or similar triangles, or trigonometry, etc.)
- differentiate the equation with respect to t implicitly to get an equation connecting rates
- substitute your data and solve
3.11 Linear Approximations and Differentials (T, C)
- know what is meant by the linear approximation L(x) to a function f(x) at a point and how to use it in approximating the function
- what is the differential of a function and how do you compute it
- understand the meaning of the differential of a function as an error in calculating f(x + dx) -f(x) when dx is small
4.1 Maximum and Minimum Values (C)
- understand the definitions, and distinctions between, relative(local) and absolute (global) maximum and minimum values of a function
- know Fermat's theorem and what is meant by a critical point of a function
- be able to find critical points and hence determine the local maxima and minima of a function.
- understand how to find the absolute maximum and minimum values a function defined on a finite interval
4.3 How Derivatives Affect the Shape of a Graph and 4.5 Summary of Curve Sketching (T,C)
- know the basic connections between the formula for the derivative and
- regions where the function is increasing and decreasing
- 1st derivative test for local max and local min
- know the connection between the second derivative and curvature
- concave up and concave down are determined by the sign of the second derivative
- inflection points are points where the concavity changes.
- second derivative test for local max and local min
- keep in mind the check list of things to ask yourself when sketching a curve
4.4 Indeterminate Forms and L'Hopital's Rule (C)
- be able to recognize any of the indeterminate forms
- know how to compute indeterminate form limits using L'Hopital's rule (or transform your problem into a form where this rule can be applied)
4.7 Optimization Problems (C)
- know the main steps in setting up and solving optimization word problems
- draw a picture
- choose variables and label the picture
- write down any restrictions your variables have to satisfy
- write down a formula for what you want to optimize
- eliminate siome variables using the restrictions above to write your quantity as a function of a single variable
- take a derivative, set to zero, etc.
4.10 AntiDerivatives (T)
- know what is meant by an anti-derivative of a function
- be familiar with the fact that two antderivatives of the same function must differ by a constant, so this provides a way to find "all" antiderivatives of a function
- be familiar with the use of "extra" conditions to evaluate the constant in an antiderivative
5.1 Areas and Distances (G, T)
- know how to estimate the area under a curve using Riemann approximations
- know what is mean by left-hand, right-hand, and mid-point approximations and how to compute them in specific cases
- be familiar with the connection between finding areas and finding a distance function when you are given a velocity function
5.2 The Definite Integral (T)
- know the definition of the definite integral as a limit of a Riemann sum
- understand the concept of signed areas
- know properties 1 thru 8 of the definite integral and how to use them in simplifying calculations
5.3 The Fundamental Theorem of Calculus (T, G, C)
- know the two different parts of the fundamental theorem and how they are useful
- Part 1: if a function g is defined as a definite integral from (any constant) to x, then the derivative of g is the integrand evaluated at x
- Part 2: the value of an integral from a to b is the value of an antiderivative of the integrand at b minus its corresponding value at a
5.4 Indefinite Integrals and the Total Change Theorem (T, C)
- know what is meant by the indefinite integral of a function: the general antiderivative of the integrand. In particular, it must contain an arbitrary additive constant.
- know the list of indefinite integrals given on page 402 as well as the two rules (at the top of the table) for finding indefinite integrals of complicated functions in terms of simpler functions
5.5 The Substitution Rule (C)
- know how to simplify an integral by making a substitution u = g(x). This involves two steps:
- write down du = g'(x)dx and use it as a way of eliminating dx in exchange for du
- solve u = g(x) for x, say x = h(y), and use this as a way of eliminating x in exchange for u
- this will leave an integral in u alone to evaluate
- know what to look for in deciding upon an appropriate substitution. This is mainly the internal part of a composite function in the integrand in question
- when you evaluate definite integrals by substitution, take the x integration from a to b and change it to a u integration from c = g(a) to d = g(b)
6.1 Areas Between Curves (C)
- know the "strip" method for computing areas between curves. The specifics of this method might involve:
- finding the intersection points of the two curves so that the limits of integration can be found
- viewing the area in terms of strips of width dy parallel to the x-axis
- dividing the integration interval into a number of sub-intervals depending on the sign of f1(x) - f2(x)
6.2 Volumes (C)
- know the methods for using integrals to compute volumes. These invariably involve "slicing" the volume into disks that are either parallel to the x-axis or the y-axis and adding up the volumes of the disks (using an integral)
- be able to apply the disk methods to volumes of revolution
6.4 Work (C)
- know how to compute the work done by a variable force, especially for springs
- know how to find the work done in tank-emptying problems using the method of slices
6.5 Average Value of a Function (C)
- know how to compute the average value of a function on an interval
8.3 Fluid Pressure - Centroids (C)
- be able to apply the slice method to finding the force on walls due to fluid pressure
- know the formulas for the mass and the components of the centroid of a region and be able to apply them in simple problems