Calculus I Final Exam
Suggestions for Review
The following list is meant to supplement your studies for the final. The
list is not exhaustive in terms of material to review. It is meant as a
guide and includes common areas of struggle in each section. This was compiled
from the individual exam guides, with some changes and additions for the
rest of Chapter 6.
General Comments
50% or more of the questions on the final will
involve material from Chapter 6. (Please keep in mind that most concepts
from Chapters 1 through 4 can be tested using functions from Chapter 6.)
The rest will be on chapters
1--4. You should understand the preliminary material well, but we will
ask about Chapter P only as it relates to material in later chapters.
!!!STUDY BY DOING LOTS OF HOMEWORK PROBLEMS!!!
CHAPTER 1
Section 1.1
-
Know the average rate of change and slope of secant line (p.52-53).
-
Understand the informal definition of limit (p.55).
-
Be sure to simplify an expression by doing the necessary algebra
before finding the limit. For example, cancel any common factor
from the numerator and denominator of a rational function before you calculate
the limit.
-
Review Examples 8 and 9 (p.56-57).
Section 1.2
-
Know the properties of limits and Example 1 (p.61-62).
-
Know how to eliminate zero denominators algebraically and Examples 4 and 5
(p.64).
-
Understand the Sandwich theorem and Example 7 (p.64).
Section 1.3
-
Understand but do not memorize the formal definition of limit
(p.70).
-
Skip the material on finding deltas for given epsilons.
-
You do not need to know how to prove the theorems.
Section 1.4
-
Review the various definitions for left-hand, right-hand, and infinite
limits. You do not need to memorize the definitions.
-
Know that a rational function with
no common factors has infinite
limits as x approaches the zeros of the denominator. Review Examples 4-6
(p.81-82).
Section 1.5
-
Remember the definition of continuity (p.87) and the continuity test
(p.89).
-
Know the following: for a removable discontinuity at x = c, f(x)
must
have a limit L as x approaches c, but f(c) is either undefined or different
from L; the discontinuity is removed by the continuous extension
of f(x) to F(x) (p.92); and the new function F(x) coincides with f(x) except
at x = c, where F(c) = L.
-
Know that for a jump discontinuity, f(x) has different one-sided
limits as x approaches c from the left and right, but for an infinite
discontinuity, f(x) has an infinite limit.
-
Review the Intermediate Value Theorem (p.93). If a continuous function
f(x) changes sign in an interval [a,b], then it must have a root
in that interval.
Section 1.6
-
Know the definition of the slope of a curve and its tangent line (p.99)
and how to calculate them.
-
Be able to find the equation of the tangent line by evaluating the tangent
slope at the given point and then using the point-slope formula for a straight
line.
-
Understand that, for a moving particle, the instantaneous speed
at t = a is the slope of the line tangent to the displacement
curve
y = f(t) at t= a.
CHAPTER 2
Section 2.1
-
Know the definition of the derivative (p.109) and how to calculate the
derivative of a given function at a given point using the definition.
Review Examples 1 and 2 (p.110-112) and the alternative formula for calculating
derivatives (p.117).
-
Know when a function does not have a derivative at a point
(p.114-115).
-
Know that differentiable functions are continuous, but continuous functions
are not necessarily differentiable.
-
Skip graphing f' from estimated values (p.112-113).
Section 2.2
-
Differentiation rules: remember them (especially the product and quotient
rules) and learn how to use them, but you do not need to prove
them.
-
Understand the evaluation of derivatives at a given point: f'(a) is the
value
of the derivative function f'(x) at x = a.
-
Understand second and higher order derivatives and how to compute them
(p.128).
Section 2.3
-
Understand average and instantaneous rates of change of a function
(p.131).
-
For the motion along a line, know how to calculate velocity, speed, and
acceleration from a given displacement function. For the difference between
speed and velocity: know that positive velocity => forward motion and negative
velocity => backward motion. Review Example 3 (p.134).
-
You do not need to memorize the free-fall equations (p.135).
-
Understand the application of rate of change in other areas, such as Economics,
Chemistry, and Biology.
Section 2.4
-
Know what the theorems of this section state and their significance.
-
Understand Examples 1, 2, and 5.
-
Know the derivatives of the six basic trig functions. (You can derive the
other four from the formulas for the derivatives of sine and cosine.)
Section 2.5
-
Understand the Chain Rule and the notation that goes along with it, and
understand how to use it, including its repeated use.
-
Learn the "Outside-Inside" rule on page 156.
Section 2.6
-
Understand how to compute derivatives by implicit differentiation.
-
Know how to use implicit differentiation to compute slopes, tangents, and
normals to curves.
-
Know how to use implicit differentiation to compute higher derivatives.
-
Understand the Power Rule for rational powers and how to use it.
Section 2.7
-
Understand and be able to use the strategy described on page 174.
-
Understand the examples.
-
Know the formulas for (i) the areas and perimeters of triangles,
rectangles, and circles and (ii) volumes and surface areas for cubes, spheres,
and right circular cylinders.
CHAPTER 3
Section 3.1
-
Make sure you clearly understand the definitions of absolute (global) and
local extreme values (maximum and minimum) and critical points.
-
Know the Max-Min and First Derivative Theorems. Remember to check
endpoints! Note that not all
critical points are necessarily local maximum or local minimum (see Example
5, page 194).
-
Know how to compute absolute extreme values of functions on closed
intervals.
Section 3.2
-
Understand Rolle's and the Mean Value Theorems and the three important
corollaries. You are responsible for the statement and meaning of
these theorems, but not their proofs.
-
Be able to apply these theorems and corollaries: be able to find a function
from knowing its derivative, including being able to find velocity and
position from knowing acceleration, and know how to determine where a function
is increasing or decreasing.
Section 3.3
-
Be able to apply the First Derivative Test for local extreme values.
-
Understand the examples. Use the sign pattern of f'(x) over intervals to
determine where f(x) is increasing and decreasing.
Section 3.4
-
Know the definitions of concave up, concave down, point of inflection,
and cusps. Note that not all points where f''(x) is either zero
or does not exist are necessarily inflection points (see Example 5, page
212).
-
Know and be able to apply the two Second Derivative Tests, one for concavity
and one for local extreme values.
-
Use the sign pattern of f'' to determine where f(x) is concave up or concave
down.
-
Skip graphing a function using its first and second derivatives.
Section 3.5
-
Know the definitions of the limit as x approaches plus or minus infinity
and the properties in Theorem 6.
-
Understand how to take limits of rational functions as x approaches plus
or minus infinity; remember while you study this that it is best to think
in terms of 1/x.
-
Know the definitions of horizontal, vertical, and oblique asymptotes and
how to compute them.
-
Skip graphing a function using its asymptotes and dominant terms.
Section 3.6
-
You need not remember the specifics of any of the examples or theorems
in this section, but you should be able to do similar problems using the
strategy described on page 236.
Section 3.7
-
Know the definitions of linearization and differentials.
-
Be able to compute the linearization of a function.
-
Understand the relationship between derivatives and differentials and how
to estimate absolute, relative, and percentage change using differentials.
Section 3.8
-
Understand Newton's method: know the formula and the geometric interpretation,
and be able to apply it.
-
Skip the last three subsections on convergence, things that go wrong,
and chaos.
CHAPTER 4
Section 4.1
-
Know the definitions of antiderivative and indefinite integral, as well
as the notation.
-
Know the formulas in Table 4.1 (p.277).
-
Know the rules for indefinite integration on page 278, and how to use
them.
-
You should know trig identity (2) on page 41. You do not need
to know the other trig identities like the angle sum and double angle formulas
(p.42) and the law of cosines (p.43). If you need them, we will supply them to
you.
Section 4.2
-
Know how to find a function F(x) given its derivative f(x) and its value
y=F(a) at some point x=a.
-
Be able to do problems similar to the examples, such as finding position
x(t) given acceleration a(t), initial position x(0), and initial velocity
v(0).
-
Skip Sketching Solution Curves, Mathematical Modeling, and Computer
Simulation.
Section 4.3
-
Know how to run the Chain Rule backward: be able to find antiderivatives
by substitution.
-
Know the Power Rule for antiderivatives and remember the specific formulas
for sin, cos, sec2, csc2, secˇtan, and
cscˇcot
(table at the bottom of page 293).
Section 4.4
Section 4.5
-
Understand the notation and terminology for summations (Sigma notation),
Riemann sums, and definite integrals. You must know the algebra rules on
page 310, but not the sum formulas on page 311.
-
Know the definition of the definite integral as a limit of Riemann sums
when it exists--and that this means as ||P|| goes to zero and for any choice
of ck. You do not need to know the precise epsilon-delta
definition on page 313.
-
Know Theorem 1 and what it means, and how the definite integral relates
to area under the curve for a nonnegative function as defined on page 318.
-
Know how to do problems like Example 9 on page 318 (you will be given
the relevant sum formulas on page 311 if necessary).
Section 4.6
-
Know all the rules for definite integrals on page 324 and how to use them.
You do not need to know their proofs.
-
Know how to compute area between a curve and the x-axis using the
definite integral.
-
Know the definition on page 328 of the average of a function. Know the
Mean Value Theorem for integrals, but not its proof.
Section 4.7
-
Know both parts of the Fundamental Theorem, what they mean, and how to
use them. These are the two most important theorems in in Calculus I! You must
understand both parts, but you do not need to know their proofs.
-
Know how to take derivatives of definite integrals whose limits of integration
are functions of x. Review Examples 1-3 on pages 334-335.
-
Know how to evaluate a definite integral using an indefinite integral;
review Examples 4 and 5 on pages 336-337.
Section 4.8
-
Know how to use substitution in definite integrals. You can use either
of the methods outlined on page 343, but you must consistently follow
only one method in a given problem.
Section 4.9
-
Know how to make a uniform partition of [a,b] into n subintervals by setting
h=(b-a)/n and xk=a+kh, k=0,1,2,...,n.
-
Know how to apply the Trapezoidal Rule and Simpson's Rule to estimate definite
integrals.
-
You do not need to memorize the formulas for these rules and their
errors. We will supply them if needed--but you should understand
them and know how to apply them.
CHAPTER 6
Section 6.1
-
Know the definition and significance of what it means for a function to
be one-to-one, and know the horizontal line test on page 450.
-
Know: (i) the definition of the inverse of a function; (ii) when the inverse
exists; (iii) how the domain and range of the inverse function are related
to the domain and range of the original function; and (iv) how to compute
the inverse of a function.
-
Know how to compute the derivative of the inverse of a function directly
or using Theorem 1. Be sure you understand from Theorem 1 that the derivatives
are to be evaluated at two different places (a and f-1(a)).
You can use either form of the derivative of the inverse function i.e.,
(f-1 )'(a) = 1/f'(f-1(a)) or (f-1 )'(f(a))
= 1/f'(a).
Section 6.2
-
Know the definition of ln(x), its derivative, and its range and domain.
-
Learn how to apply the properties of natural logarithms given in Table
6.1. Review Examples 3 and 4 on page 460. You do not need to know
the proofs of these properties.
-
Know what logarithmic differentiation is and how to use it. Review the
steps outlined on page 463.
-
Know what the integral of 1/u is and how to use it, as in Examples 6 and
7 on page 464. Know the integrals of tan and cot.
Section 6.3
-
Know the definition of ex, its relationships with ln(x), its
derivative and integral, and its range and domain.
-
Learn how to apply the properties of ex given in Table 6.2,
especially how to remove logarithms or exponentials from an equation. Review
Examples 1-3 on page 468-9. You do not need to know the proofs of
these properties.
Section 6.4
-
Know the definitions of ax and loga(x), their
relationships
with each other ((4) and (5) on page 478), the laws that govern them (Tables
6.3 and 6.4), their derivatives, and the integral of au ((3)
on page 477).
-
Review the examples.
Section 6.5
-
Know the Law of Exponential Change and how to apply it to growth and decay.
You should know how to solve for y(t) from the equation y' = ky with a
given initial value y(0) = yo.
-
You should be familiar with the examples of this section and how to do
similar problems, but you do not need to memorize them.
Section 6.6
-
Know both forms of L'Hopital's Rule and how to apply them.
-
Know how to write intermediate forms using algebra so that these theorems
can be applied. Review the examples.
Section 6.7
-
Understand Relative Rates of Growth and the concepts of order, o(g), and
O(g). Skip the subsection Sequential vs. Binary Search.
-
Review the examples.
Section 6.8
-
Know the restrictions on the domains of the basic trig functions that make
them 1-1 (see the box on page 505). Understand what this means about the
existence of their inverses and know the definitions on pages 506-7.
-
Be able to compute common values of the inverse function (those that relate
to pi/4, pi/6, and pi/3) and be able to work with the standard identities
(see (4)-(8)).
-
Review the examples and know how to use triangles to obtain angles in a
given quadrant.
Section 6.9
-
Be familiar with the derivatives of the inverse functions (Table 6.5) and
related integrals (Table 6.6), but do
not memorize them. We will
supply them to you if necessary.
-
Be familiar with the examples, especially those involving completing the
square.