Calculus I, Math 170, Kriloff
Objectives
Section 1.1
- Know the definition of function and be able to work with
functions given as graphs, equations, tables, and in words.
- Be able to find the domain of a function and sketch its graph.
- Be able to work with and construct piecewise functions.
- Be able to construct functions described in words.
- Understand the graphical symmetries of odd and even functions and
be able to test for them algebraically.
Section 1.2
- Know the basic families of functions (polynomials, roots,
rational, trigonometric, exponential, and logarithmic functions),
properties of each, and how to work with them.
- Know the difference between algebraic and transcendental
functions.
- Have a basic understanding of the relationship between
mathematical models and conclusions and real world problems and
predictions.
Section 1.3
- Understand the algebraic and graphical effects of vertical and
horizontal shifts, stretches, and reflections of functions.
- Know how to combine functions by arithmetic operations and by
composition
Section 1.4
- Know how mathematical understanding can help you use a graphing
calculator effectively.
- Be aware of pitfalls that can occur when using a graphing
calculator without sufficient mathematical understanding.
Section 2.1
- Know the definition of a secant line and the definition of a
tangent line.
- Understand that we use the limit of the slopes of secant lines to
find the slope of a tangent line.
- Be able to compute slopes of secant lines and estimate slopes of
tangent lines from tables of data.
- Know the definitions of average and instantaneous velocity and
their relation to slopes of secant and tangent lines.
Section 2.2
- Know the notation and rough definition for the one and two-sided
limits of a function f(x) as x approaches a.
- Be able to read limits from graphs and guess at limits from
values of a function found using a calculator.
- Understand how infinite limits relate to asymptotes.
- Be able to analyze behavior of functions to determine infinite
limits.
Section 2.3
- Understand the statements, especially hypotheses, of the limit
laws and their consequences.
- Be able to use the limit laws, possibly together with algebra, to
evaluate limits.
- Be able to apply the Squeeze Theorem, including checking
hypotheses, to evaluate limits.
- Know why the Squeeze Theorem will be important later.
Section 2.4
- Know that there exists a precise, rigorous definition of the
limit of f(x) as x approaches a.
- Understand the statement of the precise definition of the limit
of f(x) as x approaches a.
- Be able to estimate from a graph a value of delta for a specific
epsilon in the definition of limit.
- (Certain 300 level math courses, assume you are familiar with the
rigorous definition. Try proving some easy limits using it.)
Section 2.5
- Know the definition of continuity including that f(a) and the
limit of f(x) as x approaches a exist.
- Be able to find locations of continuity or discontinuity from a
graph or a formula.
- Be able to use the definition and limits to explain why a
function is continuous or discontinuous.
- Know types of functions that are continuous on their domains.
- Know the statement of the Intermediate Value Theorem and why
continuity is required.
Section 2.6
- Know the precise definition of tangent line and its slope using a
limit of a difference quotient.
- Be able to find slopes and equations of tangent lines to simple
functions using the definition.
- Know how slopes of secant and tangent lines relate to average and
instantaneous rates of change in various physical situations, including
velocities.
Section 3.1
- Know the definition of the derivative of a function f at a number
a.
- Be able to find derivatives of simple functions using the
definition.
- Be able to state the meaning of the derivative of a function f at
a number a as the slope of the tangent line to f at x=a and as the
instantaneous rate of change of f(x) with respect to x at x=a.
Section 3.2
- Understand the derivative as a function; be able to find the
derivative function from the definition.
- Be able to find or construct graphical information about f '(x)
from f(x) and about f(x) from f '(x).
- Know how f can fail to be differentiable and be able to find such
points from the graph of f.
- Know the theorem that if a function f is differentiable, then f
is continuous.
Section 3.3
- Know the basic derivative formulas and understand how they are
proven.
- Be able to find the derivatives of basic functions using the sum,
difference, product, constant multiple, and quotient rules.
- Be able to find the derivative at a point and the equation of a
tangent line using the basic rules.
Section 3.4
- Know examples of how rates of change are used in other fields.
- Be able to describe meanings and find units of rates of change in
many physical situations.
- Be able to calculate rates of change from formulas and use these
to answer practical questions.
- Be able to estimate rates of change from tables.
Section 3.5
- Understand the algebraic proofs and the geometric evidence for
the differentiation formulas for trigonometric functions.
- Know the trigonometric limits used in the proofs that d/dx(sin x)
= cos x and d/dx(cos x) = -sin x.
- Know and be able to use the formulas for the derivatives of
trigonometric functions.
- Be able to find limits involving trigonometric functions.
Section 3.6
- Understand from examples why the Chain Rule is needed to
differentiate composite functions.
- Be able to recognize composite functions and find their
derivatives using the Chain Rule primarily starting from formulas, but
also from graphs and tables.
Section 3.7
- Be able to recognize when it is appropriate to use implicit
differentiation.
- Know how to use implicit differentiation to find a derivative of
one variable (regarded as dependent) with respect to another (regarded
as independent), even when we cannot solve for the dependent variable in
terms of the independent variable.
Section 3.8
- Be able to calculate higher derivatives from formulas and be able
to identify and sketch their graphs.
- Understand the physical meanings of the second derivative of
position as acceleration and the third derivative of position as jerk.
- Be able to find formulas for the n-th derivative of basic
functions by examining patterns in the first few derivatives.
(This skill will be needed in Chapter 12 in Math 175.)
Section 3.9
- Know the strategy for solving related rates problems on p. 202.
- Be able to use the strategy to solve for the rate of change in
one quantity given information about the rate of change in another
quantity.
Section 3.10
- Understand the interpretation of the tangent line as a linear
approximation to a function. (In Chapter 12 in Math 175 you will
study higher degree polynomial approximations to functions.)
- Be able to find the linear approximation to a function.
- Be able to use the linear approximation to make predictions and
estimations in physical situations.
- Be able to find differentials and understand their geometric
interpretation on a graph (used in 5.5).
Section 4.1
- Be able to state precise definitions of local and absolute
maximum and minimum, critical value.
- Be able to identify the various extreme values and their
locations on graphs and be able to construct functions when extreme
values and related behavior are specified.
- Be able to locate critical values of a function given by a
formula.
- Know Fermat's Theorem and be able to use its logic correctly to
draw conclusions.
- Be able to use the Closed Interval Method to find absolute
minimum and maximum values of a continuous function on a closed interval
when the function is given by a formula.
Section 4.2
- Be able to test the hypotheses for Rolle's Theorem and the Mean
Value Theorem.
- Be able to use a formula or a graph to find a value c satisfying
Rolle's Theorem and the Mean Value Theorem.
- Be aware that the Mean Value Theorem will be used to prove the
Increasing/Decreasing Test.
Section 4.3
- Be able to use the Increasing/Decreasing Test to find intervals
on which a function f given by a formula is increasing or decreasing.
- Be able to use the First Derivative Test to find local maximum
and minimum values of f.
- Be able to state the definitions of points of inflection, concave
up, and concave down.
- Be able to use the Concavity Test to find intervals on which f is
concave up or down.
- Be able to use the Second Derivative Test to find local maximum
and minimum values of f.
- Know how to specify behavior of a function f when given a graph
of f, f ', or f ''.
Section 4.4
- Be able to state the rough definition for the limits of f(x) as x
approaches infinity and -infinity.
- Know how to find limits at infinity for functions given by
formulas or in graphs.
- Be able to state the definition of horizontal asymptote.
- Be able to use limits at infinity to find horizontal asymptotes
and sketch graphs of functions.
- Be aware that there are precise definitions of limits at infinity.
Section 4.5
- Be able to use techniques from algebra, such as domain,
intercepts, symmetry, and periodicity, together with techniques from
calculus, such as limits to find asymptotes, derivatives to find
intervals of increase and decrease, local extreme values, and intervals
of concavity, to sketch the graph of a function by hand.
Section 4.6
- Be able to effectively use both graphing by hand and graphing by
calculator to identify important aspects of the behavior of a function.
Section 4.7
- Be able to set up and solve optimization problems.
- Be able to clearly document and explain the steps used in solving
an optimization problem.
Section 4.9
- Be able to state what Newtion's method is used for and why it is
important.
- Be able to estimate values and predict outcomes of Newton's
method from a graph.
- Be able to perform Newton's method to estimate roots of equations.
Section 4.10
- Be able to state the definition of antiderivative.
- Be able to find the most general antiderivative of a function.
- Be able to find position from initial position and velocity or
acceleration and initial velocity.
- Be able to construct a direction field and sketch antiderivatives
using a direction field.
Section 5.1
- Be able to give over- and underestimates for the area under a
curve using rectangles, starting from a formula, a graph, or a table.
- Be able to use the behavior of a function to decide whether a
left or right hand estimate is an overestimate or an underestimate.
- Be able to write estimates using sigma notation (also see
Appendix E).
- Be able to use the area under a velocity curve to find distance
travelled.
Section 5.2
- Be able to use the definition of the definite integral to
evaluate definite integrals using left endpoints, midpoints, or right
endpoints, starting from a formula, a graph, or a table.
- Be able to set up and evaluate Riemann sums using the basic sums
and properties on page 327.
- Be able to use the properties of the definite integral on pages
331 and 333 to manipulate and compare integrals.
- Be able to interpret the definite integral as area to calculate
integrals.
Section 5.3
- Be able to state both parts of the Fundamental Theorem of
Calculus.
- Be able to use part 2 of the Fundamental Theorem to evaluate
definite integrals.
- Be able to use part 1 of the Fundamental Theorem to find
derivatives of functions defined by integrals.
- Be able to interpret the Fundamental Theorem as showing that
differentiation and integration are opposite processes.
Section 5.4
- Know that indefinite integrals are the same as antiderivatives.
- Be able to find indefinite integrals of basic functions.
- Be able to use the definite integral to find total change.
Section 5.5
- Be able to use substitution to find indefinite integrals.
- Be able to use substitution to evaluate definite integrals, using
correct notation.
- Be able to use symmetry to simplify evaluation of definite
integrals.
Mastering the substitution method will pay off dramatically in Math 175!
Section 6.1
- Be able to set up a limit of a Riemann sum representing an area
between curves and recognize it as a definite integral.
- Be able to draw regions bounded by given equations.
- Be able to find the area of a region bounded by given curves by
calculating the intersection points of the two curves and using a
definite integral.
Section 6.2
- Be able to set up a limit of a Riemann sum representing the
volume of a solid with given cross sectional area and recognize it as a
definite integral.
- Be able to calculate volumes of solids obtained by rotating a
region bounded by given curves around a horizontal or vertical line.
Section 6.3
- Be able to recognize when it is more useful to use shells than
slices to find volumes.
- Know why the formula for the volume of a solid using shells works.
- Be able to calculate volumes of solids using a definite integral
and cylindrical shells.
Section 6.4
- Know the formula for work as a definite integral and why it holds.
- Be able to calculate work using a definite integral of force.
Section 6.5
- Know the formula for the average value of a function on an
interval and why it holds.
- Be able to calculate the average value of a function on an
interval using a definite integral.