Perform the various algebraic and trigonometric computations encountered in calculus.
Demonstrate understanding of the concept of the limit of a function – from both a graphical perspective and an algebraic perspective – and the role of the limit concept in defining continuity.
Find derivatives by using the basic definition involving a limit.
Find derivatives of polynomial, rational, radical, and trigonometric functions, as well as composites of such functions, by using the power rule, the product rule, the quotient rule, the chain rule, and implicit differentiation, as appropriate.
Use both the prime notation and the Leibniz notation for derivatives.
Use first and second derivatives, together with algebraic analysis, to produce an accurate graph of a given function, identifying all asymptotes, symmetry, relative maximums and minimums, stationary and inflection points, and intervals on which the function is increasing, decreasing, concave up, and concave down.
Use derivatives to solve problems that involve related rates.
Use differentials, when appropriate, in problem-solving.
Evaluate definite integrals by using the basic definition involving the limit of a Riemann sum.
Evaluate integrals, using substitution where appropriate.
Use integration to solve simple differential equations.
Use integration to find areas of plane regions, lengths of plane curves, surface areas of solids of revolution, and volumes of such solids by disk, washer, and shell methods.
Use calculus in applications involving rectilinear motion, moments and centroids of plane regions, fluid force, and work.