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Instructions on the exam

“Leave your answer in calculator-ready form / find the exact value”: Don't round. Stop when only button-pressing is left, like $\sqrt{40}$ or $2 + \dfrac{\log 3}{5}$ , and keep $\pi$ , $e$ , roots and $\log$ as symbols.
“Simplify your answer”: Finish the algebra: one fraction, common factors canceled, numbers combined.
“Remember to completely justify your answer”: Show why, not just what: the sign chart, the endpoint check or the limit that proves it. A bare number earns little.
“You do not need to provide justification”: Only the final answer is marked here, so answer and move on. Save your time for the long questions.
“Write "None" if there is no such interval / or clearly state that there are none”: An empty answer is still an answer: write None instead of leaving the blank empty.
“Give units to your answer / treat units properly”: A rate's units are output units per input unit: dollars per day, km per minute, radians per minute.Related Rates →
“Use the limit definition of the derivative. Any answer not using the limit definition will receive 0 marks”: Start from the difference quotient and simplify until $h$ cancels. Shortcut rules like the power rule score zero here. $f'(x) = \lim_{h\to 0}\dfrac{f(x+h) - f(x)}{h}$ Derivative Definition →
“Put your answer in the form $y = mx + b$ ”: Write the line in point-slope form first, then expand and solve for $y$ .Tangent Lines →

Slopes, tangent lines and approximation

“Find the equation of the tangent line to the curve at the point $(a, b)$ ”: Point-slope form: the slope is $f'(a)$ and the point is $(a, f(a))$ . The derivative alone is not the answer; it is only the slope. $y - f(a) = f'(a)(x - a)$ Tangent Lines →
“What is the slope of the graph of $y = f(x)$ at $x = 3$ ?”: Just a number: find $f'(x)$ , then plug in $x = 3$ . $f'(3)$ Tangent Lines →
“the tangent line(s) parallel to the line $y = \tfrac{1}{2}x - 1$ ”: Parallel means equal slopes: solve $f'(x) = \tfrac{1}{2}$ , then build a tangent line at each solution. $f'(x) = \tfrac{1}{2}$ Tangent Lines →
“Find all points where the tangent is horizontal”: Slope zero: solve $f'(x) = 0$ . On an implicit curve, set the top of $\dfrac{dy}{dx}$ to $0$ (bottom not $0$ ) and solve it with the curve's equation. $f'(x) = 0$ Tangent Lines →
“the tangent line is vertical”: On $y = f(x)$ : $f$ is defined at the point but $f'(x)$ blows up to $\pm\infty$ there (often at an endpoint of the domain). On an implicit curve: the bottom of $\dfrac{dy}{dx}$ is $0$ while the top is not; solve that together with the curve's equation.Implicit Differentiation →
“the linear approximation of $f(x)$ at $x = a$ / use the tangent line to approximate”: It is exactly the equation of the tangent line at $x = a$ (solved for $y$ ), standing in for $f$ near $a$ : build $L(x)$ at an easy nearby point, then plug in. For $\sqrt{1.04}$ , use $\sqrt{x}$ at $a = 1$ . $L(x) = f(a) + f'(a)(x - a)$ Tangent Lines →
“the third-degree Taylor polynomial for $f(x)$ about $x = a$ ”: Add up the terms below for $k = 0$ to $3$ . A Maclaurin polynomial is the same thing with $a = 0$ . $T_3(x) = \sum_{k=0}^{3}\dfrac{f^{(k)}(a)}{k!}(x - a)^k$ Taylor & Newton →
“Use two iterations of Newton's method, starting at $x_0 = 3$ ”: Apply the update twice to get $x_1$ , then $x_2$ . To approximate a number like $\sqrt{8}$ , first make it a root: $f(x) = x^2 - 8$ . $x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$ Taylor & Newton →

Computing derivatives

“Variables $x$ and $y$ are related by the equation … Find $\dfrac{dy}{dx}$ . Your answer may depend on both $x$ and $y$ ”: Implicit differentiation: differentiate both sides in $x$ , give every $y$ -term a factor $y'$ (chain rule), then solve for $y'$ . Leaving $y$ in is fine. $\dfrac{d}{dx}\,y^3 = 3y^2\,y'$ Implicit Differentiation →
“Find $\dfrac{dy}{dx}$ in terms of $x$ ”: Only $x$ may appear. After logarithmic differentiation gives $y' = y\cdot(\ldots)$ , replace $y$ by its formula.Log Differentiation →
“Write your result in the form $f'(x) = f(x)\,G(x)$ / find $\dfrac{h'(x)}{h(x)}$ ”: Logarithmic differentiation: $G$ is the derivative of $\log f$ . Expand the log first, so products and quotients become sums. $G(x) = \dfrac{f'(x)}{f(x)} = \dfrac{d}{dx}\log f(x)$ Log Differentiation →
“Identify the inside function and the outside function”: In $\cos(x^3 + 2)$ the inside is $x^3 + 2$ and the outside is $\cos$ . Chain rule: the outside's derivative at the inside, times the inside's derivative. $\dfrac{d}{dx}\,f(g(x)) = f'(g(x))\,g'(x)$ Derivative Rules →
“functions defined for all real numbers, with derivatives of all orders”: You may differentiate as often as you need ( $f'$ , $f''$ , $f'''$ , …), and every derivative is continuous, so its limit at a point is its value there.Derivative Rules →

Rates and motion

“the average velocity of the particle from $t = 1$ to $t = 1.001$ ”: No derivative: change in position over change in time. $\dfrac{s(1.001) - s(1)}{0.001}$ Rates of Change →
“How fast is … changing (or decreasing) when …?”: A rate at one moment: relate the quantities by an equation, differentiate in $t$ , then plug in that moment's numbers. For "decreasing", give the size of the negative rate. $\dfrac{dQ}{dt}\Big|_{\text{that moment}}$ Related Rates →
“production is kept constant”: That quantity does not change, so its rate is $0$ . $\dfrac{dP}{dt} = 0$ Related Rates →
“the angle of inclination $\theta$ is increasing at 0.1 radians per minute”: The angle up from the ground to the object, so $\tan\theta = \dfrac{\text{height}}{\text{ground distance}}$ . Keep $\theta$ in radians. $\dfrac{d\theta}{dt} = 0.1$ Related Rates →
“the lamp rotates at 1 revolution per minute”: Convert to radians: one revolution is $2\pi$ radians. $\dfrac{d\theta}{dt} = 2\pi \text{ rad/min}$ Related Rates →
“the rate of change of $E$ per unit length (i.e. $\dfrac{dE}{dL}$ )”: $E$ and $L$ both depend on a third variable $a$ , so divide their rates. $\dfrac{dE}{dL} = \dfrac{dE/da}{dL/da}$ Related Rates →

Limits and long-run behavior

“Evaluate the following limit or show it does not exist”: Try to compute it. Only if it truly has no value (the one-sided limits differ, or it oscillates) say so, and show why.Limits →
“Do not use l'Hôpital's rule. State the leading-order behavior of the numerator and denominator”: Swap each piece for its small- $x$ approximation, like $\sin u \approx u$ , $e^u - 1 \approx u$ , $\log(1+u) \approx u$ , $1 - \cos u \approx \tfrac{u^2}{2}$ , then cancel the powers of $x$ .Limits →
“Explain whether the exponential or polynomial terms dominate”: As $x \to +\infty$ , $e^x$ beats every power of $x$ . As $x \to -\infty$ , $e^x \to 0$ , so the powers win. $\log x \ll x^p \ll e^x \quad (p > 0,\ x \to +\infty)$ Limits at ∞ →
“the best approximation of $F(t)$ as $t \to \infty$ / $s(x) \sim a(x)$ as $x \to -\infty$ ”: Keep only the dominant term on top and on the bottom, and simplify that ratio. The $\sim$ means the ratio tends to $1$ . $\lim \dfrac{s(x)}{a(x)} = 1$ Limits at ∞ →
“What power function best approximates f(x) near the origin? For "large" x?”: Near the origin keep the lowest power on top and on the bottom; for large $x$ keep the highest. The answer looks like $Cx^p$ .Limits at ∞ →
“Find all vertical and horizontal asymptotes. If there is no horizontal asymptote, or no vertical asymptote, explain why”: $0$ Asymptotes →

Domain, continuity and differentiability

“Find the domain / give the interval(s) on which f is continuous”: Throw out every $x$ that makes a denominator $0$ , the inside of an even root negative, or the inside of a log $\le 0$ . What is left is the answer.Continuity →
“list all x-values where the pictured function has a jump discontinuity”: Both one-sided limits are finite but different: the graph jumps. Holes and vertical asymptotes do not count. $\lim_{x\to a^-} f(x) \ne \lim_{x\to a^+} f(x)$ Continuity →
“Find the two values of the constant $a$ that make the function continuous at $x = 3$ ”: Plug $x = 3$ into both pieces, set the two values equal and solve for $a$ . A quadratic in $a$ is why there can be two. $\lim_{x\to 3^-} f(x) = \lim_{x\to 3^+} f(x) = f(3)$ Continuity →
“Find the value of $a$ such that the function has no vertical asymptote at $x = 4$ ”: The bottom is $0$ at $x = 4$ . Pick $a$ so the top is $0$ at $x = 4$ too; then $(x - 4)$ cancels and there is no wall there.Asymptotes →
“Is f differentiable at x = 2? If so determine f′(2); if not explain why not”: If $f$ is built from differentiable pieces (no zero denominator, no corner) at $x = 2$ , say so and compute $f'(2)$ with the rules. If not, show the one-sided slopes differ.Derivative Definition →
“Find the constants $a$ and $b$ that make $F$ differentiable at $x = 1$ . Explain why checking continuity alone is not enough”: Two equations for two unknowns: the pieces' values match at $x = 1$ , and so do their slopes. A continuous graph can still have a corner. $F_1(1) = F_2(1),\quad F_1'(1) = F_2'(1)$ Derivative Definition →

Graphs, max and min

“determine the intervals where f(x) is increasing and the intervals where f(x) is decreasing (or concave up / concave down)”: Sign charts: $f' > 0$ increasing, $f' < 0$ decreasing; $f'' > 0$ concave up, $f'' < 0$ concave down.Curve Sketching →
“Find all inflection points”: Points where $f''$ changes sign. Solving $f''(x) = 0$ only gives candidates, so check the sign on each side.Curve Sketching →
“the sketched graph is the derivative of f(x), and not f(x) itself”: Read $f$ from it: above the axis, $f$ increases; where the sketch rises, $f$ is concave up. Its zeros are $f$ 's critical points.Curve Sketching →
“Draw a large sketch, clearly labeling the x-values of any asymptotes, extrema and inflection points”: The marks are for the features: write each asymptote, max, min and inflection $x$ -value on the graph, and make the shape match your sign charts.Curve Sketching →
“Identify every critical point and every singular point”: When a question names both, a critical point has $f'(x) = 0$ and a singular point is where $f'(x)$ does not exist (elsewhere both are just called critical points). Local extrema can only occur at these (or at endpoints).Curve Sketching →
“Find the global maximum and minimum of the function on the interval $[-3, 3]$ ”: Evaluate $f$ at each critical point inside the interval and at both endpoints. The biggest value is the max, the smallest the min.Optimization →
“A closed tank is to have a volume of … Find the radius and height that minimize its total surface area”: The quantity after "minimize" is your function; the fixed amount (a volume, a budget of coins) is the constraint that cuts it down to one variable.Optimization →
“Explain, without differentiation, why f must have a global maximum”: Use values and limits, not $f'$ : a continuous $f$ that climbs above its starting value and then sinks back as $x \to \infty$ must reach a highest point.Theorems →

What the question is asking

Instructions on the exam

Slopes, tangent lines and approximation

Computing derivatives

Rates and motion

Limits and long-run behavior

Domain, continuity and differentiability

Graphs, max and min

Which topics are you reviewing?

Navigation

Instructions on the exam

Slopes, tangent lines and approximation

Computing derivatives

Rates and motion

Limits and long-run behavior

Domain, continuity and differentiability

Graphs, max and min