Course 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).