Calculus and Discrete Mathematics
This domain blends introductory calculus-limits, continuity, derivatives, and basic integration concepts-with discrete topics: sequences and series, the binomial theorem, combinatorics, and logic. You should understand limits intuitively and algebraically, interpret the derivative as a rate of change and slope of the tangent line, and recognize arithmetic versus geometric sequences.
Traps to watch
- Limits at removable discontinuities: a factor that cancels means the limit exists even though the function is undefined there.
- Derivative as slope, not value: $f'(a)$ is the instantaneous rate at $a$; don't confuse it with $f(a)$.
- Arithmetic vs. geometric: arithmetic adds a constant difference; geometric multiplies by a constant ratio. Their sum formulas differ.
- Counting order: permutations count arrangements (order matters), combinations count selections (order does not).
Worked example / rule of thumb
Evaluate $\displaystyle\lim_{x\to 2}\frac{x^2-4}{x-2}$. Factor: $\frac{(x-2)(x+2)}{x-2} = x+2$ for $x\neq 2$. As $x\to 2$, the limit is $4$. Rule of thumb: for $0/0$ limits, factor and cancel before substituting. For sequences, identify the type first: a geometric series with $|r|<1$ converges to $\frac{a}{1-r}$. When a problem asks "how many ways," decide whether order matters before choosing a formula.