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Instructions on the exam

“Leave your answer in calculator-ready form”: Don't round. Write the exact sum or expression, and stop when only button-pressing is left. Keep $\pi$ , $e$ , roots and $\log$ as symbols.
“Write down a definite integral that represents the area … Do not evaluate the integral explicitly.”: The set-up is the whole answer: the right limits and the right . Stop once the integral is written.
“Express … as a definite integral. Do not evaluate this integral.”: Match the sum to $\sum f(x_i)\,\Delta x$ : find $\Delta x$ , $a$ , $b$ and $f$ , and stop at the integral.Definite Integral →
“Is the following reasoning sound? If not, fix it.”: In a substitution, check two things: was $dx$ converted to $du$ with the right constant, and were the limits changed to $u$ -values?Substitution →
“Be sure to say why your method can be applied to this particular series”: State the test's conditions. For the alternating series estimate: the signs alternate and $b_n$ decreases to $0$ .Convergence Tests →
“Use the comparison test (not the limit comparison test)”: Write an explicit inequality, $a_n \leq b_n$ or $a_n \geq b_n$ , that holds for all large $n$ . Don't use a limit of $\dfrac{a_n}{b_n}$ here.Convergence Tests →

Definite integrals and the FTC

“Evaluate the following integral by interpreting it as a signed area, and using geometry”: Sketch the graph, cut it into triangles, rectangles and circle pieces, and count area below the axis as negative. No antiderivatives.Definite Integral →
“Using a limit of right-endpoint Riemann sums, evaluate …”: Write $R_n$ with $x_i = a + i\,\Delta x$ , use the summation formulas, and take $n \to \infty$ . An antiderivative is only for checking. $\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(a + i\,\Delta x)\,\Delta x$ Definite Integral →
“Find the best possible upper estimate”: Use the endpoints that give the taller rectangles: right endpoints if the function is increasing, left if it is decreasing.Definite Integral →
“Evaluate the following integrals using the Fundamental Theorem of Calculus Part 2, or explain why it does not apply”: Check that the integrand is continuous on all of $[a, b]$ . If it is not, the theorem does not apply: say where it breaks.Fundamental Theorem →
“What is the instantaneous rate of change of $F(x)$ with respect to $x$ ?”: Find $F'(x)$ . For $F(x) = \int_a^x f(t)\,dt$ , that is just $f(x)$ . $F'(x) = f(x)$ Fundamental Theorem →

Techniques of integration

“Derive the reduction formula”: Do one round of integration by parts with a general $n$ . Then write the result using the same integral with a smaller power. $\int u\,dv = uv - \int v\,du$ Integration by Parts →
“Choose a trigonometric substitution that will eliminate the roots”: Complete the square if needed, then sine for $a^2 - u^2$ , tangent for $a^2 + u^2$ , secant for $u^2 - a^2$ .Trig Substitution →
“Your answer may not contain inverse trigonometric functions”: After a trig substitution, read $\sin\theta$ , $\cos\theta$ or $\tan\theta$ off a right triangle instead of writing $\theta = \arcsin(\dots)$ .Trig Substitution →
“You may use that $\int\sec x\,dx = \log|\sec x + \tan x| + C$ ”: A hint that the integral becomes $\int\sec\theta\,d\theta$ , usually after $x = a\tan\theta$ or $x = a\sec\theta$ .Trig Substitution →
“Write out the general form of the partial-fractions decomposition. You need not determine the values of any of the coefficients”: Just the template: one term per linear factor, every power for a repeated factor, $Bx + D$ over each .Partial Fractions →
“Find the coefficient of $\dfrac{1}{x - 1}$ in the partial fraction decomposition”: You need only one constant. Clear denominators and plug in $x = 1$ . If $x - 1$ is repeated, compare coefficients instead.Partial Fractions →
“Re-write as the sum of a polynomial and a rational function whose numerator has a strictly smaller degree than its denominator”: Use long division. The answer is the quotient plus the remainder over the denominator.Partial Fractions →
“Factor … into linear and irreducible factors”: Factor until each quadratic left has $b^2 - 4ac < 0$ . Those stay as they are.Partial Fractions →

Numerical integration

“Find a reasonable value $M$ such that $|f''(x)| \le M$ ”: Any number that is truly at least $|f''(x)|$ for every $x$ in the interval. It does not have to be the maximum. A smaller valid $M$ just gives a tighter error bound.Numerical Integration →
“How many subintervals are required in order to guarantee that the absolute error is less than …”: Set the error bound $\le$ the tolerance, solve for $n$ , and round up (to an even number for Simpson's rule).Numerical Integration →
“Give the relative error, absolute error, and percent error”: Absolute error $= |\text{exact} - \text{approx}|$ . Relative error $=$ absolute error $\div\ |\text{exact}|$ . Percent error $= 100 \times$ relative error.Numerical Integration →

Improper integrals

“Evaluate the integral … or state that it diverges”: Find every bad point: an infinite limit, or a point of $[a, b]$ where the integrand blows up. Split there and take a limit on each piece. If every piece has a finite limit, add them. If any piece does not, the integral diverges. $\int_a^\infty f\,dx = \lim_{R\to\infty}\int_a^R f\,dx$ Improper Integrals →
“Does the integral … converge or diverge? Justify your claim”: You don't have to find the value. Compare with $\int \frac{dx}{x^p}$ or $\int e^{-x}\,dx$ , and name the comparison test you used.Improper Integrals →
“For which values of $b$ is the integral … improper?”: Improper means a limit of integration is $\pm\infty$ , or the integrand blows up somewhere on the interval. List every $b$ that makes either happen.Improper Integrals →

Area, volume and averages

“the finite region bounded by the curves”: The enclosed piece between the curves. Its limits are where the curves meet: solve $f(x) = g(x)$ .Area Between Curves →
“Find the total area between the curves … on the interval …”: Count every piece as positive: split where the curves cross and integrate top minus bottom on each piece. $\int_a^b |f(x) - g(x)|\,dx$ Area Between Curves →
“rotated about the line $y = -1$ ”: Measure every radius from the line $y = -1$ , not from the $x$ -axis: radius $= y - (-1) = y + 1$ .Volumes →
“Using vertical slices (disks and/or washers)”: Slice with thickness $dx$ , so the axis must be horizontal. Each slice is $\pi r^2\,dx$ or $\pi(R^2 - r^2)\,dx$ .Volumes →
“The cross-sections perpendicular to the $x$ -axis are squares with one side in the $xy$ -plane”: Slice with thickness $dx$ . The square's side is the width of the base at $x$ , so $A(x) = (\text{top} - \text{bottom})^2$ . $V = \int_a^b A(x)\,dx$ Volumes →
“Calculate the root mean square of $f(x)$ over $[a, b]$ ”: Square $f$ , average the square over $[a, b]$ , then take the square root. $\sqrt{\dfrac{1}{b - a}\int_a^b f(x)^2\,dx}$ Averages →

Work and center of mass

“How much work does it take to pump out all the water?”: Slice the water horizontally. Each slice weighs $\rho g\,A(y)\,dy$ and is lifted to the top or the spout. Integrate weight times distance over the water. $W = \int \rho g\,A(y)\,(\text{distance})\,dy$ Work →
“A force of … N is required to hold a spring stretched … cm beyond its natural length”: This gives $k$ in Hooke's law $F = kx$ , with $x$ in meters: $6$ N at $3$ cm gives $k = \dfrac{6}{0.03} = 200$ N/m. $F = kx$ Work →
“Find the centroid of the finite region bounded by …”: The center of mass with constant density. Find the area, then $\bar x$ and $\bar y$ . $\bar x = \dfrac{1}{A}\int_a^b x(T - B)\,dx,\quad \bar y = \dfrac{1}{2A}\int_a^b \left(T^2 - B^2\right)dx$ Center of Mass →
“Using symmetry, find the centroid”: The centroid lies on every line of symmetry. Two lines of symmetry pin it down with no integrals.Center of Mass →
“Assume it has constant density $\rho$ ”: $\rho$ cancels from top and bottom, so the center of mass is the centroid.Center of Mass →

Differential equations

“Express your solution explicitly as $y = y(x)$ ”: Solve for $y$ : no $y$ may remain on the right side.Separable Equations →
“You don't have to solve for $y$ in terms of $x$ ”: An answer is enough: $G(y) = F(x) + C$ with $C$ found from the initial condition.Separable Equations →
“decreasing at a rate proportional to the square of the velocity”: "Proportional to" means a constant $k > 0$ times it. "Decreasing" puts in the minus sign. $\dfrac{dv}{dt} = -kv^2$ Separable Equations →
“What is the limiting value of $P$ as $t$ becomes large?”: Solve for $P(t)$ , then take $\displaystyle\lim_{t \to \infty} P(t)$ .Separable Equations →

Sequences

“State whether this sequence converges or diverges, and if it converges give its limit”: Converges means the limit is a finite number. Going to $\pm\infty$ or bouncing around both count as diverging.Sequences →
“choose an upper bounding sequence and lower bounding sequence that will work with the squeeze theorem”: Find $b_n \le a_n \le c_n$ with $b_n$ and $c_n$ going to the same limit.Sequences →
“satisfies the recursion relation $a_{n+1} = \ldots$ ”: Each term is built from the one before. If $a_{n+1} = f(a_n)$ with $f$ continuous, any limit $L$ solves $L = f(L)$ . Then show the gap $a_n - L$ goes to $0$ .Sequences →

Series and convergence

“Evaluate the following series, or show that it diverges”: First decide whether it converges. If it does, it is almost always geometric or . Then find the exact sum.Series →
“Show that the series … converges and find its limit”: Find $\lim_{N\to\infty} S_N$ . It is usually telescoping: write out $S_N$ and cancel.Series →
“The $n$ th partial sum of a series is known to have the formula $s_n = \dots$ ”: $a_1 = s_1$ , and $a_n = s_n - s_{n-1}$ for $n \geq 2$ . The sum of the series is $\lim s_n$ .Series →
“Express … as a rational number, i.e. in the form $p/q$ where $p$ and $q$ are integers”: The number is a repeating decimal. Write its repeating part as a geometric series and sum it. Then add the part that does not repeat.Series →
“Determine whether the series … is absolutely convergent, conditionally convergent, or divergent”: Test $\sum |a_n|$ first. If it converges, the series is absolutely convergent. If not, try the alternating series test: passing it means conditionally convergent.Convergence Tests →
“According to the Alternating Series Estimation Theorem, what is the smallest value of $N$ …”: Check that $b_n$ decreases to $0$ , then solve $b_{N+1} \leq$ tolerance for the smallest $N$ . $|S - S_N| \leq b_{N+1}$ Convergence Tests →

Power series and Taylor series

“Find the radius of convergence and interval of convergence of the series”: The ratio test with $x$ kept in gives $|x - a| < R$ . Then plug in each endpoint and test it separately.Power Series →
“Find a power series representation for …”: Rewrite as $\dfrac{A}{1 - r}$ and use the geometric series, or differentiate or integrate a series you know. $\dfrac{1}{1 - r} = \sum_{n=0}^{\infty} r^{n}$ Power Series →
“Using the definition of a Taylor series, find the Taylor series for … centered at …”: Find a formula for $f^{(n)}(a)$ and build the series from it. Don't just substitute into a known series here. $\sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x - a)^{n}$ Taylor Series →
“Express the Taylor series … in summation notation”: Give the general term inside a $\sum$ , not just the first few terms.Taylor Series →
“Give the first two nonzero terms in the Maclaurin series”: Skip the terms that are $0$ and write the first two that are not, like $x - 2x^{5}$ .Taylor Series →
“Use series to evaluate $\displaystyle\lim_{x \to 0}$ …”: Replace each function by its Taylor series, cancel, and read off the lowest power of $x$ that survives.Taylor Series →

What MATH 101 Exam Questions Are Really Asking

Instructions on the exam

Definite integrals and the FTC

Techniques of integration

Numerical integration

Improper integrals

Area, volume and averages

Work and center of mass

Differential equations

Sequences

Series and convergence

Power series and Taylor series

Which topics are you reviewing?

Navigation

Instructions on the exam

Definite integrals and the FTC

Techniques of integration

Numerical integration

Improper integrals

Area, volume and averages

Work and center of mass

Differential equations

Sequences

Series and convergence

Power series and Taylor series