Final review: limits, derivatives, applications and integrals

This page is for the last read before the final. I have gathered the most frequently asked part of each week in one place; for details go back to the page of the relevant week. The final exam usually consists of four blocks: limits and continuity, computing derivatives, applications of derivatives, and integrals. At least one question comes from each block, and the application questions (related rates, optimization, curve sketching) carry the most points.

Study suggestion: read the tables below, then before looking at the questions, write your own solution on paper. When you open the answer, look not only at the result but at the order of the steps. In the exam, partial credit comes from the steps.

Block 1: limits and continuity (weeks 3-4)

In a continuity question write the three conditions explicitly: $f(a)$ defined, the limit exists, they are equal. In a “removable” discontinuity the limit exists; in a “jump” the left and right differ; in an “infinite” one the limit is infinite.

Block 2: computing derivatives (weeks 5-7)

Derivative from the definition: $f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$. In a fraction find a common denominator, in a root multiply by the conjugate, cancel $h$.

For derivatives that look hard, the order is: first simplify (expand with log rules, divide the fraction), then look at the outer structure (product, quotient or composition), chain rule last. If the base and exponent are both variable, use logarithmic differentiation. If $y$ cannot be written explicitly, use implicit differentiation: every term with $y$ is multiplied by $y'$.

Derivative of an inverse function: $(f^{-1})'(b)=\dfrac{1}{f'(a)}$, $f(a)=b$. Find $a$ first.

Block 3: applications of derivatives (weeks 8-10)

Related rates. Figure, variables, equation, derivative with respect to $t$, then the numbers at that moment. Don’t fix a changing quantity before differentiating.

Absolute extrema (closed interval): critical points ($f'=0$ or undefined) and endpoints; table of values; the largest and the smallest.

MVT: $f'(c)=\dfrac{f(b)-f(a)}{b-a}$. For an “exactly one root” question: at least one by the IVT, at most one by Rolle.

Curve sketching: domain, intercepts, asymptotes, $f'$ sign table (increasing/decreasing, local extrema), $f''$ sign table (concavity, inflection points). Without a sign change there is no extremum or inflection point.

Optimization: objective function + constraint, reduce to one variable, write the interval, $f'=0$, show that it is a max/min.

Linear approximation: $L(x)=f(a)+f'(a)(x-a)$, with $a$ a nearby point whose value is known. Error: $dQ=Q'\,dx$, relative error $\dfrac{dQ}{Q}$.

Block 4: integrals (weeks 11-14)

Riemann sum: $\Delta x=\dfrac{b-a}{n}$, $x_i=a+i\Delta x$, sigma formulas, limit. The definite integral is signed area.

Fundamental Theorem: $\dfrac{d}{dx}\displaystyle\int_a^{g(x)}f(t)\,dt=f(g(x))g'(x)$ and $\displaystyle\int_a^bf=F(b)-F(a)$.

When you see an integral, try this in order:

1. Directly from the table?       (first expand, divide, simplify)
2. Inner function + its derivative?           -> u-substitution
3. Product of two different types of function? -> integration by parts (LIATE)
4. Polynomial / polynomial?                   -> degree check, partial fractions
5. a^2 - x^2, a^2 + x^2, x^2 - a^2 present?   -> trigonometric substitution
6. ax^2 + bx + c present?                     -> complete the square first

$+C$ in an indefinite integral, an absolute value in a $\ln$ answer, converting the limits when substituting in a definite integral. These three are where points are lost most often.

Time and score management in the exam

• Finish the easy blocks first. Limit and direct derivative questions are two-to-three-minute jobs; starting with them both secures points and warms up the mind.
• In word problems, write the figure and the variable definitions on paper. Even if the result comes out wrong, a correctly set up equation and a correct derivative earn partial credit.
• Check every antiderivative by differentiating. It takes 20 seconds and catches most sign errors.
• Check the units and re-read the question. Is “maximum value” asked, or “where”? Velocity or magnitude of velocity? If it says “exact value”, leave $\ln 2$, $\pi$, $\sqrt3$ instead of a decimal.
• Leave a question you are stuck on and come back. Losing fifteen minutes on a single integral means losing two easy questions.

There is also a quick list for testing yourself: can you recite each of the following from memory? $\lim\frac{\sin x}{x}$, the product and quotient rules, the six trigonometric derivatives, the derivatives of $e^x$, $\ln x$, $\arctan x$, $\arcsin x$, the closed interval method, the formula for $L(x)$, the two parts of the Fundamental Theorem, the formula for $\int u\,dv$, the three trigonometric substitutions. Re-read the week of any item you can’t recite.

Common mistakes (whole semester)