MATH 31A flash cards
48 cards for Differential and Integral Calculus, laid out week by week and timed around your exams. Free on iPhone and the web.
Exam dates
Unblank times your reviews around these. You can always set your own dates in the app.
- FinalWed, Mar 17, 2027A sample date until the Registrar posts the real one
The first 20 cards
Read the question, think of your answer, then open the card.
Week 1 · Limits and one-sided limitsWhat does lim x→a f(x) = L mean?Show the answerHide the answer
f(x) gets as close to L as you like once x is close enough to a, with x ≠ a.
The value f(a) plays no part. It can differ from L or not exist at all.
Week 1 · Limits and one-sided limitsWhat is lim x→0 of |x|/x?Show the answerHide the answer
It does not exist. |x|/x is −1 for every x < 0 and 1 for every x > 0, so the one-sided limits disagree.
Week 1 · Limits and one-sided limitsWhat is lim x→1⁺ of 1/(x − 1)?Show the answerHide the answer
∞. Just right of 1, x − 1 is a tiny positive number, so its reciprocal grows without bound.
From the left the limit is −∞, so x = 1 is a vertical asymptote.
Week 1 · Limits and one-sided limitsWhat is lim x→2 of (x² − 4)/(x − 2)?Show the answerHide the answer
4. Factor the top as (x − 2)(x + 2), cancel x − 2, then put in x = 2.
Week 2 · Computing limitsWhat is lim x→0 of sin(5x)/x?Show the answerHide the answer
5. Write it as 5 · sin(5x)/(5x), and sin u / u → 1 as u → 0.
Week 2 · Computing limitsWhat is lim x→∞ of (3x² + 1)/(5x² − x)?Show the answerHide the answer
3/5. Divide top and bottom by x²; every other term goes to 0.
When top and bottom have the same degree, the limit at infinity is the ratio of the leading coefficients.
Week 2 · Computing limitsHow does the squeeze theorem give lim x→0 of x² sin(1/x)?Show the answerHide the answer
The limit is 0. Since −x² ≤ x² sin(1/x) ≤ x² and both bounds go to 0, the middle must too.
Week 2 · ContinuityWhat three conditions make f continuous at a?Show the answerHide the answer
f(a) is defined, lim x→a f(x) exists, and the two are equal.
Week 2 · Continuityf(x) = kx for x < 2 and x² + 1 for x ≥ 2. What k makes f continuous?Show the answerHide the answer
k = 5/2. The pieces must meet at x = 2, so 2k = 2² + 1 = 5.
Week 2 · ContinuityWhy must x³ + x − 1 = 0 have a solution between 0 and 1?Show the answerHide the answer
The polynomial is continuous, −1 at x = 0 and 1 at x = 1, so by the intermediate value theorem it takes the value 0 in between.
Week 3 · The derivative as a limitHow is f′(a) defined as a limit?Show the answerHide the answer
f′(a) = lim h→0 of (f(a + h) − f(a))/h, the slope of the tangent line at x = a.
Week 3 · The derivative as a limitUsing the limit definition, what is the derivative of x²?Show the answerHide the answer
2x. ((x + h)² − x²)/h simplifies to 2x + h, which goes to 2x as h → 0.
Week 3 · The derivative as a limitWhy is |x| not differentiable at 0?Show the answerHide the answer
Its slope is −1 left of 0 and 1 right of 0, so the difference quotient has no limit there.
|x| is still continuous at 0. Differentiable implies continuous, but continuous does not imply differentiable.
Week 3 · Power rule and tangent linesWhat is the derivative of 4x³ − 2√x?Show the answerHide the answer
12x² − 1/√x, by the power rule, since √x = x^(1/2) has derivative 1/(2√x).
Week 3 · Power rule and tangent linesWhat is the tangent line to y = x³ at x = 1?Show the answerHide the answer
y = 3x − 2. The slope is 3x² = 3 at x = 1, and the line passes through (1, 1).
Week 4 · Product and quotient rulesWhat is the derivative of x² sin x?Show the answerHide the answer
2x sin x + x² cos x, by the product rule (fg)′ = f′g + fg′.
Week 4 · Product and quotient rulesWhat is the derivative of x/(x + 1)?Show the answerHide the answer
1/(x + 1)². The quotient rule gives ((x + 1) · 1 − x · 1)/(x + 1)².
Week 4 · Derivatives of trig functionsWhat is the derivative of tan x?Show the answerHide the answer
sec² x. Treat tan x as sin x / cos x; the quotient rule's numerator is cos² x + sin² x = 1.
Week 4 · Derivatives of trig functionsWhat is the 99th derivative of sin x?Show the answerHide the answer
−cos x. The derivatives cycle every 4 (cos x, −sin x, −cos x, sin x), and 99 leaves remainder 3.
Week 5 · The chain ruleWhat does the chain rule say?Show the answerHide the answer
The derivative of f(g(x)) is f′(g(x)) · g′(x): the outer derivative at the inside, times the inner derivative.
Week by week
The whole deck: 48 cards over 10 weeks. Every week is open from day one.
- 1Limits4 cards
- Limits and one-sided limits 4
- 2Computing limits and continuity6 cards
- Computing limits 3
- Continuity 3
- 3The derivative5 cards
- The derivative as a limit 3
- Power rule and tangent lines 2
- 4Product, quotient and trig rules4 cards
- Product and quotient rules 2
- Derivatives of trig functions 2
- 5Chain rule and implicit differentiation5 cards
- The chain rule 3
- Implicit differentiation 2
- 6Related rates and linear approximation4 cards
- Related rates 3
- Linear approximation 1
- 7Extreme values and the mean value theorem5 cards
- Extreme values 2
- The mean value theorem 3
- 8Shape of a graph and L'Hôpital's rule5 cards
- Concavity and the derivative tests 3
- L'Hôpital's rule 2
- 9Optimization and antiderivatives4 cards
- Optimization 1
- Antiderivatives 3
- 10Definite integrals and the fundamental theorem6 cards
- Riemann sums and definite integrals 2
- The fundamental theorem of calculus 4