Calculus and Pre-Calculus
What this domain tests
Calculus and Pre-Calculus in Math 521A bridges function analysis and introductory calculus: limits, continuity, average and instantaneous rates of change, derivative rules, and basic applications such as tangents, extrema, and motion. Pre-calculus content includes polynomial and rational behavior, sequences and series, and trigonometric function analysis.
Core rules and concepts
- Limits: evaluate by direct substitution first; if 0/0 results, factor, rationalize, or simplify. A limit exists only if left- and right-hand limits agree.
- Continuity at x = a requires f(a) defined, the limit existing, and the two being equal.
- Difference quotient: average rate of change is [f(a+h) − f(a)]/h; the derivative is its limit as h → 0.
- Derivative rules: power rule d/dx xⁿ = nxⁿ⁻¹; product, quotient, and chain rules; d/dx sin x = cos x, d/dx cos x = −sin x, d/dx eˣ = eˣ, d/dx ln x = 1/x.
- Applications: f′(x) = 0 locates critical points; sign changes classify maxima and minima; f″ indicates concavity.
- Sequences and series: arithmetic adds a constant difference, geometric multiplies by a constant ratio; the geometric sum converges only when |r| < 1, to a/(1 − r).
Worked example
Find the equation of the tangent line to f(x) = x³ − 2x at x = 1.
First, the point: f(1) = 1 − 2 = −1, so the point is (1, −1). Next, the slope: f′(x) = 3x² − 2, so f′(1) = 3 − 2 = 1. Then use point-slope form: y − (−1) = 1(x − 1), giving y = x − 2.
Check: at x = 1, y = −1, matching the function value. The derivative supplies slope, the function supplies the point - both are needed.
Traps that cost marks
- Forgetting the chain rule's inner derivative when differentiating composites such as sin(3x) or (2x + 1)⁵.
- Sign errors on the derivative of cosine.
- Confusing average rate of change (a secant slope) with instantaneous rate (a tangent slope).
- Using the geometric series formula when |r| ≥ 1, where the sum diverges.
- Declaring a critical point an extremum without checking the sign change.