# Course topics

### Basics of differentiation

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

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

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).