MATH 100 practice

Power functions and first sketches

Which term dominates for small and large x, sketching two-term polynomials, and the familiar functions.

Functions, graphs and inverses

Domain, composition, transformations, inverses and the trigonometry the course assumes.

Limits

Limit laws, the indeterminate form 0/0, one-sided and infinite limits, limits at infinity and asymptotes.

Continuity

The definition of continuity, classifying discontinuities, the constants that make a piecewise function continuous, and the Intermediate Value Theorem.

The definition of the derivative

Secants and tangents, average and instantaneous velocity, the difference quotient and differentiability.

Differentiation rules

Power, sum, product and quotient rules, tangent lines and higher-order derivatives.

Exponential and trigonometric functions

Derivatives of eˣ, aˣ, sine, cosine and tangent, and the limits behind them.

The chain rule and logarithms

Compositions, nested rules, the natural logarithm and logarithmic differentiation.

Implicit differentiation and inverse trigonometric functions

Differentiating relations, tangent lines to curves, and the derivatives of inverse and inverse trigonometric functions.

Related rates

Translating a moving situation into an equation, then differentiating.

L’Hôpital’s rule

Indeterminate forms 0/0, ∞/∞, 0·∞, ∞−∞, 1^∞, 0⁰ and ∞⁰, and when the rule does not apply.

Curve sketching

Increase, concavity, inflection points, asymptotes and full sketches.

Optimization

Building an objective function and justifying the extreme value.

Linear approximation

Tangent-line estimates, whether they run high or low, and approximating a value without a calculator.

Taylor polynomials

Quadratic and higher approximations, the Maclaurin polynomials of the standard functions and, for flavour A, bounds on the error.

Newton’s method

Following the tangent line towards a root: an iteration or two by hand, and the starting points where it fails.

Flavours A and B only

Exponential growth and decay

The differential equations y′ = ky and y′ = a − by: half-lives, doubling times, Newton’s law of cooling and inflow and outflow.

Flavours A and B only

Slope fields and Euler’s method

Checking that a function solves a differential equation, reading a slope field, and stepping a solution forward with Euler’s method.

Flavours A and B only

Logistic growth and phase lines

Steady states and their stability from a state-space diagram (phase line), the logistic equation, and harvesting.

Flavour B only

Functions of two variables and partial derivatives

Evaluating and reading functions of two variables, level curves, partial derivatives as marginal rates, and second partials.

Flavour C only

Optimization in two variables and Lagrange multipliers

Critical points and the second derivative test, the extremes of a function on a closed region, and Lagrange multipliers.

Flavour C only

The Mean Value Theorem

Rolle’s theorem, the MVT, counting roots and proving inequalities.

Antiderivatives

Reversing differentiation, the constant of integration and initial conditions.