## Topics

### Basics of differentiation

• Find average rate of change and connections to instantaneous rate of change.
• Understand limits; rules for evaluating limits; one-sided limits; limits involving infinity; asymptotes.
• The definition of continuity and how to use/identify continuous functions. Be able to recognize different types of discontinuities (both in terms of functions and graphs).
• Give the limit definition of a derivative at a point and as a function (when possible) and use it to calculate the derivative of simple functions. Give units of derivative (when needed).
• How to compute higher order derivatives. Know and use the various symbols for derivatives.
• Find tangent lines to function and interpret tangent lines geometrically as local approximation to function.
• The sum, product, quotient, and chain rules for differentiation.
• The derivatives of x^k, e^x, sin(x), cos(x), tan(x), sec(x).
• Differentiation as it relates to position, velocity, and acceleration.

### Advanced techniques and applications of differentiation

• Implicit differentiation and derivatives of inverse functions.
• The derivatives of ln|x|, arctan(x), arcsin(x), arcsec(x).
• Use logarithmic differentiation.
• Use linearization (differentials) to approximate a function locally.
• Set up and solve related rates problems.
• Understand absolute max/min and local (or relative) max/min.
• Know and understand the mean value theorem for derivatives.
• Identify and classify critical points by use of either the first derivative test or the second derivative test.
• Identify when a function is increasing or decreasing on a given interval.
• Identify when a function is concave up or concave down on a given interval; and find points of inflection.
• Be able to sketch a function using information from critical points and the first and second derivatives.
• Use L'Hospital's rule to work with indeterminant forms (0/0 or ∞/∞). Use tools to take other ambiguous limits to an appropriate indeterminant form.
• Set up and solve optimization problems; including justifying that the solution is the extreme value sought and give correct units.
• Use Newton's method to approximate roots of functions.

### Basics of integration

• Find antiderivatives of basic functions including x^k, e^x, sin(x), cos(x), sec^2(x), sec(x)tan(x), 1/(1+x^2), sec(x), and tan(x). Know that antiderivatives are unique up to "+C".
• Use Riemann sums to approximate net area under a curve. Know how to read sum notation and manipulate sums using algebraic rules.
• Give relationship between Riemann sums and definite integral. Know basic properties of definite integrals (constant multiple, sums, inequalities, breaking into multiple parts, reversing order).
• Compute average of a function over an interval using definite integrals.
• Understand the mean value theorem for definite integrals.
• State and use both variations of the Fundamental Theorem of Calculus.
• Use substitution for definite and indefinite integration.
• Find the net area under a curve, as well as area between two curves.
• Integration as it relates to acceleration, velocity, and position.
• (*)Solve separable differential equations (with and without initial conditions).

(*) = Topic covered during dead week; not on Exam 3