Calculus I: study guide and practice
Also known as: math 151, math 2413, math 1550, math 1910, math 140, calc 1, calc i, calculus 1, calculus i, differential calculus, calculus for science and engineering, math 121
What you will be able to do
- Estimate and evaluate limits, including one-sided limits, and decide where a function is continuous
- Define the derivative as a limit and interpret it as a tangent slope and an instantaneous rate of change
- Differentiate polynomials, products, quotients and composite functions with the standard rules
- Differentiate trigonometric, exponential and logarithmic functions
- Solve related-rates and optimisation problems and sketch curves using the first and second derivatives
- Evaluate definite and indefinite integrals and interpret them as signed area and net change
- Apply the fundamental theorem of calculus and integrate by substitution
Topics
- LimitsA limit describes what a function approaches near a point. Learn to estimate limits, use one-sided limits and evaluate them with algebra.
- Continuity and the limit lawsContinuity means no break in the graph. Learn the limit laws, types of discontinuity, the intermediate value theorem and the squeeze theorem.
- The derivative from the definitionThe derivative is the limit of the difference quotient. Learn tangent slopes, instantaneous rates and when a derivative fails to exist.
- Differentiation rulesMaster the power, product, quotient and chain rules, and use them for tangent slopes, velocities and second derivatives.
- Derivatives of trig, exponential and log functionsDifferentiate sin, cos, tan, e^x, a^x and logarithms, and combine them with the product and chain rules.
- Applications of derivativesUse derivatives for related rates, optimisation, curve sketching, linear approximation and the mean value theorem.
- Definite and indefinite integralsFind antiderivatives, interpret definite integrals as signed area, use Riemann sums and apply integrals to area and motion.
- The fundamental theorem and substitutionLink derivatives and integrals with the fundamental theorem, then use substitution to integrate composite functions.