Calculus I - MATH 021L
2014-01-08: This course is now over. PDF material has been removed so links below to these files will be broken.
- A Mathematica review was given December 3, 2013.
You've made it, congratulations! Indefinite integrals. Applications of integrals.
The net change theorem Integration by substitution (aka u-substitution aka change
of variables). Good luck on the final, study hard!
Proof of FTC1 Introduction and proof of FTC2. Examples using the FTC to solve integrals.
Introduction to Mathematica.
Welcome back from break, we're in the home stretch. Properties of definite
integrals. Geometric shortcuts for evaluating definite integrals. The
Fundamental Theorem of Calculus (FTC), part I.
Terminology for the definite integral. Definite integrals for non-positive
functions. Evaluating the Riemann sum and the definite integral.
The area problem and the distance problem. Definition of the definite
Review/Postmortem of Exam 2.
Chapter 5 - integral calculus! Finding the area under a curve using
More antiderivatives. Rectilinear (straight line) motion.
Quiz 8 review. Newton's Method.
Applying L'Hospital's Rule to indeterminate products, differences, and powers by rewriting
Second derivatives and the shape of a graph. Indeterminate forms and L'Hospital's Rule.
Proving the Mean Value Theorem using Rolle's Theorem. How derivatives affect the
shape of a graph. Increasing/Decreasing Test. First derivative Test.
Finding maxima and minima using derivatives. Rolle's Theorem and The
Mean Value Theorem.
Absolute and local maximum and minimum values. Extreme value theorem.
Linear approximations and linearization. Estimating linearization
uncertainty using differentials.
Continuously compounded interest. Related rates.
Quiz 6 review only.
Relative growth rates in exponential growth (and decay). Population
growth and radioactive decay.
More examples of rates of change in the sciences. Start into exponential
growth and decay.
Examples of rates of change in physics and chemistry.
Derivatives of logarithmic functions. Exploiting logs to save work:
logarithmic differentiation. Proving the power rule. Caution:
lots of similar-looking functions.
Examples showing different uses of implicit differentiation.
Derivatives of inverse trigonometric functions.
Proof of the chain rule. Implicit differentiation.
The chain rule: derivatives of composite functions.
Finished Test 1 review. Derivatives of trigonometric functions.
Derivation and examples of quotient rule for derivatives. Test 1
review. No weekly quiz.
Review of derivative rules so far. Derivation and example of the
Computing derivatives of exponential functions. Exponentials are their own derivatives.
Natural exponential function is very *cough* natural.
Computing derivatives the "easy" way. Power rule for derivatives of
power functions, constant multiple rule, sum and difference rules.
Other notations for the derivative. Differentiability, when are
functions differentiable, when do they fail to be differentiable,
Understanding derivatives for quantities beyond position and time (not
just velocity). Treating the derivative as a new function of itself.
Sketching the derivative by examining the original function. The
domains of functions compared with the domains of their derivatives.
Quiz 4 review.
Derivatives and rates of change continued. Notation for derivative of a
function, examples of computing derivatives. Average rate of change of
a general quantity y with respect to another quantity x,
take limit to get instantaneous rate of change of y with respect
Derivatives and rates of change. Tangent line is limit of secant lines,
now we can take limits directly, no more tables of numbers.
Infinite limits at infinity. Examples, examples, and more examples.
Sketching a function by looking at its behavior near its roots and near
positive and negative infinity.
Finding limits at infinity, horizontal/vertical asymptotes, algebraic
tricks to avoid indeterminate forms.
Quiz 3 review. Using the IVT. Limits at infinity, horizontal asymptotes.
Polynomials and rational functions are continuous, what types of
functions are continuous, evaluating nasty limits by exploiting
continuity, continuity and function composition. The Intermediate Value
Using the IVT.
Sec 2.5: Continuity, continuous and discontinuous functions, limits of
continunous functions, continuity on an interval, functions that are
continuous on the left and on the right,
polynomials and rational
functions are continuous.
Sec 2.4: The precise definition of a limit, proving a limit, one-sided
limits, infinite limits.
Quiz 2 review, using the squeeze theorem (Sec 2.3).
Sec 2.4: The
precise definition of a limit, proving a limit.
Example limit calculations, the squeeze theorem.
Infinite limits and calculating limits quickly using limit laws.
The velocity problem. The limit of a function. When limits exist or do
not exist. One-sided limits.
Quiz 1 review, Sec. 2.1 - Tangent lines,
the velocity problem.
Sec. 1.6 - More logarithms, natural log, graphing logs. Inverse trig
Sec. 1.6 - Inverse functions and (some) logarithms. Horizontal line
test, one-to-one functions, graphing inverse functions, recipe to find
inverse functions, logarithmic function is the inverse of the
Sec. 1.5 - Exponential functions.
Sec. 1.2 - Modeling, types of functions, even/odd functions.
Welcome to Calc I, syllabus, course info, tips and advice. Sec. 1.1 -
Functions: representations, domain/range, vertical line test,
even/odd functions, piecewise functions.