Calculus reference sheet

This sheet follows the ten units of the course in order. Each unit starts with the big idea in a few sentences, then gives the results you will look up most often, with the conditions they need. When you want the full explanation, a proof or more examples, follow the lesson links under each topic.

Angles are in radians throughout. A formula holds wherever both sides are defined unless a condition says otherwise.

Limits

A limit describes the value $f(x)$ approaches as $x$ gets close to a point, whatever happens at the point itself. Derivatives, integrals and series are all limits, so the rules here come back in every later unit.

What a limit means

We write $\lim_{x\to a} f(x) = L$ when $f(x)$ can be made as close to $L$ as we like by taking $x \ne a$ close enough to $a$. The value $f(a)$ plays no part. The precise version says that for every $\varepsilon > 0$ there is a $\delta > 0$ such that for every $x$ in the domain

The left-hand limit uses only $x < a$ and the right-hand limit only $x > a$. The two-sided limit exists exactly when both one-sided limits exist and are equal. For example $\frac{\lvert x \rvert}{x}$ tends to $-1$ from the left of $0$ and to $1$ from the right, so it has no limit at $0$.

Lessons Intuitive definition, One-sided limits, Two-sided limits, The formal definition

Limit laws

If $\lim_{x\to a} f(x) = L$ and $\lim_{x\to a} g(x) = M$ with $L$ and $M$ finite, limits can be taken piece by piece. All limits below are as $x \to a$, and the same laws hold for one-sided limits and for limits at $\pm\infty$.

Lessons Limit laws

How to evaluate a limit

Try these in order.

1. Substitute. If $f$ is continuous at $a$, for example a polynomial or a rational function defined at $a$, the limit is $f(a)$.
2. Simplify. If substitution gives $\tfrac{0}{0}$, factor and cancel, or multiply by a conjugate. This is allowed because a limit only looks at $x \ne a$.
3. Squeeze. If $g(x) \le f(x) \le h(x)$ near $a$ and $g$ and $h$ have the same limit $L$, then $f$ has limit $L$ too.
4. Use a known limit from the table below, or l’Hôpital’s rule from the Applications of Differentiation unit.

For example, for $x \ne 0$ near $0$, multiplying by the conjugate gives

Lessons Limit theorems

Special limits

The first limit needs radians. The last row says logarithms grow slower than any positive power of $x$, and powers grow slower than $e^x$.

Lessons Special limits

Infinity and asymptotes

For a rational function, only the leading terms matter as $x \to \pm\infty$. With leading terms $a_n x^n$ on top and $b_m x^m$ below,

• The line $x = a$ is a vertical asymptote when $f(x) \to \pm\infty$ as $x \to a$ from at least one side.
• The line $y = L$ is a horizontal asymptote when $f(x) \to L$ as $x \to \infty$ or as $x \to -\infty$.
• The line $y = mx + b$ is an oblique asymptote when $f(x) - (mx + b) \to 0$ as $x \to \infty$ or as $x \to -\infty$. For a rational function whose top degree is one more than its bottom degree, polynomial division finds it.

Lessons Limits at infinity, Infinite limits, Asymptotes

Continuity

A continuous function has no breaks, so its value at a point is exactly what nearby values lead you to expect. That simple property is enough to prove that equations have solutions and that maximum and minimum values exist.

Continuity at a point

A function $f$ is continuous at $a$ when

That asks for three things. The value $f(a)$ is defined, the limit exists, and the two are equal. On a closed interval $[a, b]$ only one-sided limits are used at the endpoints. Polynomials, rational functions, roots, trigonometric, exponential and logarithmic functions are continuous on their domains, and so are sums, products, quotients and compositions of continuous functions.

A removable discontinuity can be filled in. Giving $\frac{\sin x}{x}$ the value $1$ at $0$ makes it continuous everywhere.

Lessons Continuous functions, Discontinuities, Graphs of continuous functions

The intermediate value and extreme value theorems

Both theorems need $f$ continuous on a closed bounded interval $[a, b]$.

• Intermediate value theorem. Every number strictly between $f(a)$ and $f(b)$ equals $f(c)$ for some $c$ in $(a, b)$. In particular, if $f(a)$ and $f(b)$ have opposite signs, $f$ has a root in $(a, b)$. The theorem promises a root, not a unique one.
• Extreme value theorem. The function $f$ takes an absolute maximum value and an absolute minimum value on $[a, b]$. On an interval that is not closed this can fail, as $f(x) = x$ on $[0, 1)$ has no maximum.

Lessons Continuity theorems, The intermediate value theorem

Uniform continuity

A function is uniformly continuous on a set when one $\delta$ works at every point at once. For every $\varepsilon > 0$ there is $\delta > 0$ such that $\lvert f(x) - f(y) \rvert < \varepsilon$ for all $x$ and $y$ in the set with $\lvert x - y \rvert < \delta$. Every continuous function on a closed bounded interval is uniformly continuous there, while $\frac{1}{x}$ on $(0, 1)$ is continuous but not uniformly continuous.

Lessons Uniform continuity

Derivatives

The derivative measures how fast a function changes at a single point. Geometrically it is the slope of the tangent line, and physically it is an instantaneous rate such as velocity.

Definition

The derivative of $f$ at $a$ is

when this limit exists and is finite. The tangent line at $(a, f(a))$ is $y = f(a) + f'(a)(x - a)$. Other notations are $\frac{dy}{dx}$ and $\frac{d}{dx} f(x)$. A differentiable function is continuous, but a continuous function need not be differentiable. The function $\lvert x \rvert$ has a corner at $0$ and no derivative there.

Lessons Velocity and tangent, Derivatives and rates of change, The derivative as a function

Rules

Here $f$ and $g$ are differentiable and $c$ is a constant.

In Leibniz notation the chain rule reads $\frac{dy}{dx} = \frac{dy}{du}\,\frac{du}{dx}$, which is how you handle a function inside a function.

Lessons Elementary rules, Product rule, Quotient rule, Chain rule

Derivatives to know

With the chain rule each row also works for an inner function $u(x)$, as in $(\sin u)' = u' \cos u$.

The hyperbolic functions are $\sinh x = \frac{e^x - e^{-x}}{2}$ and $\cosh x = \frac{e^x + e^{-x}}{2}$, and they satisfy $\cosh^2 x - \sinh^2 x = 1$.

Lessons Trigonometric, Exponential and logarithmic, Hyperbolic

Implicit, inverse and logarithmic differentiation

• Implicit differentiation. Differentiate both sides of an equation in $x$ and $y$, remembering that each $y$ term picks up a factor $y'$, then solve for $y'$. From $x^2 + y^2 = 25$ you get $2x + 2yy' = 0$, so $y' = -\frac{x}{y}$ where $y \ne 0$.
• Inverse functions. If $f$ is one-to-one and differentiable on an interval and $f'(f^{-1}(x)) \ne 0$, then

• Logarithmic differentiation. For products, quotients and variable powers, take $\ln$ first. For $y = x^x$ with $x > 0$ this gives $\ln y = x \ln x$, so $\frac{y'}{y} = \ln x + 1$ and $y' = x^x(\ln x + 1)$.
• Higher derivatives. The second derivative $f''$ is the derivative of $f'$. For a position $s(t)$ the velocity is $s'$ and the acceleration is $s''$.

Lessons Higher derivatives, Implicit differentiation, Inverse functions, Logarithmic differentiation

Applications of Differentiation

Once you can differentiate, the sign and size of $f'$ and $f''$ tell you where a function rises, falls and bends, where it is largest, and how well a straight line can stand in for it.

Finding maximum and minimum values

A critical point is a number $c$ in the domain where $f'(c) = 0$ or $f'(c)$ does not exist. By Fermat’s theorem, a local maximum or minimum at an interior point can only happen at a critical point.

Closed interval method. For $f$ continuous on $[a, b]$, compute $f$ at the critical points inside $(a, b)$ and at $a$ and $b$. The largest value is the absolute maximum and the smallest is the absolute minimum.

To classify a critical point $c$ where $f$ is continuous,

Lessons Extreme values, The closed interval method

The mean value theorem

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then for some $c$ in $(a, b)$

At some moment the instantaneous rate equals the average rate. Rolle’s theorem is the case $f(a) = f(b)$. The consequences are used everywhere. On an interval, $f' > 0$ makes $f$ increasing, $f' < 0$ makes it decreasing, and $f' = 0$ makes it constant.

Lessons The mean value theorem, Increasing and decreasing functions

The shape of a graph

Where $f'' > 0$ the graph is concave up and bends upward, and where $f'' < 0$ it is concave down. An inflection point is a point where $f$ is continuous and the concavity changes. A zero of $f''$ is only a candidate, since $x^4$ has $f''(0) = 0$ and no inflection point.

To sketch a graph, work through the domain, intercepts and symmetry, then the asymptotes, then the sign of $f'$ for rise, fall and extrema, then the sign of $f''$ for concavity, and finally join the points in a way that agrees with all of it.

Lessons Concavity and inflection points, Sketching graphs

Rates, optimization and growth

• Related rates. Write one equation linking the changing quantities, differentiate it with respect to time, and only then put in the values at the moment you care about.
• Optimization. Write the quantity as a function of one variable using the constraint, find its domain, and use the closed interval method or the derivative tests. Check that the answer makes sense in the original problem.
• Growth and decay. The equation $y' = ky$ with $y(0) = y_0$ has the solution $y = y_0 e^{kt}$. The doubling time for $k > 0$ is $\frac{\ln 2}{k}$, and the half-life for $k < 0$ is $\frac{\ln 2}{\lvert k \rvert}$.
• Rates in the sciences. Velocity, reaction rate, population growth rate and marginal cost are all derivatives. The marginal cost $C'(x)$ approximates the cost of one more unit.

Lessons Related rates, Optimization, Rates in the sciences, Growth and decay

L’Hôpital’s rule

If $f(x)/g(x)$ has the form $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ as $x \to a$, where $f$ and $g$ are differentiable near $a$ with $g'(x) \ne 0$ there, then

provided the limit on the right exists or is $\pm\infty$. It also works for one-sided limits and for $a = \pm\infty$. Check the form before every use. Rewrite $0 \cdot \infty$ as a quotient, $\infty - \infty$ as one fraction, and the powers $1^\infty$, $0^0$ and $\infty^0$ by taking logarithms first.

Lessons Indeterminate forms and l’Hôpital’s rule

Linear approximation

Near $a$ a differentiable function is close to its tangent line,

For example $\sqrt{4.1} \approx 2 + \frac{1}{4}(0.1) = 2.025$. If $\lvert f'' \rvert \le K$ between $a$ and $x$, the error is at most $\frac{K}{2}(x - a)^2$. In the language of differentials, $dy = f'(x)\,dx$ approximates the actual change $\Delta y$.

Lessons Linear approximations and differentials

Integrals

Integration adds up infinitely many small pieces, such as thin rectangles under a curve. The fundamental theorem of calculus connects it to differentiation, so most integrals are computed by finding an antiderivative.

Antiderivatives

A function $F$ is an antiderivative of $f$ on an interval when $F' = f$ there. Any two antiderivatives on the same interval differ by a constant, which is why each entry below gets a $+\,C$. You can always check an answer by differentiating it.

Lessons Indefinite integrals, Integration rules

The definite integral

Split $[a, b]$ into $n$ pieces of width $\Delta x = \frac{b - a}{n}$ and pick a point $x_i^*$ in each. For a continuous $f$, or a bounded one with finitely many discontinuities,

When $f \ge 0$ this is the area under the graph. In general it is the area above the axis minus the area below. Integrals are linear, they add over adjacent intervals, reversing the limits changes the sign, and for $a \le b$ a larger function never has a smaller integral.

Lessons Sigma notation, The area problem, Riemann sums, Definite integrals, Properties

The fundamental theorem of calculus

Let $f$ be continuous on $[a, b]$.

Part 1. The accumulation function $g(x) = \int_a^x f(t)\,dt$ is differentiable on $(a, b)$ with $g'(x) = f(x)$. With a differentiable upper limit $u(x)$ that stays in $[a, b]$, the chain rule gives

Part 2. If $F$ is any antiderivative of $f$ on $[a, b]$, then

Read the other way, integrating a rate of change gives the net change. For a velocity $v(t)$ the integral of $v$ is the displacement and the integral of $\lvert v \rvert$ is the distance travelled.

Lessons Part 1, Part 2 and net change

Choosing a technique

Look at the integrand and pick the first row that fits.

In a definite integral, change the limits along with the variable. For example, with $u = x^2$,

For integration by parts, choose $u$ to be the factor that gets simpler when differentiated. With $u = x$ and $dv = e^x\,dx$,

The identities used most often are

Some functions, such as $e^{-x^2}$ and $\frac{\sin x}{x}$, have no elementary antiderivative at all. Their integrals come from series or numerical methods.

Lessons Substitution, By parts, Rational functions, Trigonometric integrals, Trigonometric substitutions, Hyperbolic and half angle, Other techniques

Improper integrals

An integral over an infinite interval, or of a function that blows up, is defined as a limit. For $f$ continuous on $[a, \infty)$,

and it converges when this limit is finite. If both ends are a problem, or there is a problem inside the interval, split the integral and require each piece to converge. Two families serve as yardsticks. The integral

converges exactly when $p > 1$, and the integral of the same function over $(0, 1]$ converges exactly when $p < 1$. To decide convergence without computing, compare. If $0 \le g \le f$ and the integral of $f$ converges, so does the integral of $g$.

Lessons Unbounded intervals, Singularities and comparison

Applications

Every application comes from the same idea. Cut the quantity into thin slices, approximate each slice, and add them up with an integral. The functions here are continuous.

Lessons Area between curves, Average value, Disks and washers, Shells, Arc length, Surface area, Centres of mass, Work and force, Economics and biology, Probability

Parametric Equations and Polar Coordinates

Not every curve is the graph of a function. Conics, curves traced by a moving point and curves described by angle and distance need their own descriptions, and calculus works for each of them.

Conics

Parabolas, ellipses and hyperbolas are the curves where a plane cuts a double cone. In standard position,

The eccentricity $e$ tells them apart. It is $e < 1$ for an ellipse, $e = 1$ for a parabola and $e > 1$ for a hyperbola, and $e = \frac{c}{a}$ for the ellipse and hyperbola above. A general equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ that describes a nondegenerate conic is an ellipse when $B^2 - 4AC < 0$, a parabola when $B^2 - 4AC = 0$ and a hyperbola when $B^2 - 4AC > 0$.

Lessons Definition of a conic, Parabolas, Ellipses, Hyperbolas, General conics

Parametric curves

A parametric curve gives both coordinates in terms of a parameter, $x = f(t)$ and $y = g(t)$, so it also has a direction of travel. For example $x = \cos t$, $y = \sin t$ for $0 \le t \le 2\pi$ traces the unit circle once counterclockwise. With $f'$ and $g'$ continuous on $[\alpha, \beta]$,

The tangent is horizontal where $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \ne 0$, and vertical where $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \ne 0$.

Lessons Parametric curves, Sketching, Slope, Area, Length and surfaces

Polar coordinates

A point with polar coordinates $(r, \theta)$ lies at distance $r$ from the origin along the ray at angle $\theta$. The two systems are linked by

The same point has many polar coordinates, since $(r, \theta + 2\pi)$ and $(-r, \theta + \pi)$ describe it too. Some curves are much simpler in polar form, such as the circle $r = a$, the cardioid $r = a(1 + \cos\theta)$, and the rose $r = a\cos n\theta$, which has $n$ petals for odd $n$ and $2n$ for even $n$. A conic with a focus at the origin, eccentricity $e$ and directrix $x = d$ with $d > 0$ is

Lessons Polar coordinates, Some polar curves, Sketching, Polar conics

Calculus in polar coordinates

For $r = f(\theta)$ with $f'$ continuous, and $r' = \frac{dr}{d\theta}$,

For example the cardioid $r = 1 + \cos\theta$ encloses the area

Lessons Slope, Area, Arc length

Infinite Sequences and Series

Adding infinitely many numbers can give a finite total, as $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1$ does. This unit decides when that happens and then uses it to write functions such as $e^x$ and $\sin x$ as infinite polynomials.

Sequences

A sequence $(a_n)$ converges to $L$ when its terms get and stay as close to $L$ as we like. Precisely, for every $\varepsilon > 0$ there is $N$ with $\lvert a_n - L \rvert < \varepsilon$ for all $n > N$. The limit laws and the squeeze theorem work for sequences too, and two facts are especially useful.

• If $a_n = f(n)$ and $f(x) \to L$ as $x \to \infty$, then $a_n \to L$, so l’Hôpital’s rule applies to sequences.
• A sequence that is bounded and monotonic converges.

Lessons Definition of a sequence, Convergence, Sequence theorems

Series to know

A series $\sum a_n$ converges to $S$ when its partial sums $s_N = a_1 + \cdots + a_N$ approach $S$. If it converges, then $a_n \to 0$. The converse is false, as the harmonic series shows.

Lessons Partial sums, Geometric series, Telescoping and harmonic, Series theorems

Choosing a convergence test

Work down the table and use the first test that fits.

The ratio and root tests say nothing when $L = 1$. A series converges absolutely when $\sum \lvert a_n \rvert$ converges, and absolute convergence implies convergence. A series that converges but not absolutely converges conditionally, and its terms can be rearranged to change the sum.

Lessons Integral test, Comparison tests, Alternating series, Absolute and conditional convergence, Ratio and root tests

How close is a partial sum

To approximate a sum by a partial sum $s_N$, you need a bound on the remainder $R_N = S - s_N$.

• Alternating series. If $b_n$ decreases to $0$, then $\lvert R_N \rvert \le b_{N+1}$, the first term left out.
• Integral test. Under its hypotheses, $\int_{N+1}^\infty f(x)\,dx \le R_N \le \int_N^\infty f(x)\,dx$.
• Geometric tail. If $\lvert a_{n+1} \rvert \le r\lvert a_n \rvert$ for all $n > N$ with $r < 1$, then $\lvert R_N \rvert \le \frac{\lvert a_{N+1} \rvert}{1 - r}$.

Lessons Estimates of sums

Power series

A power series $\sum_{n=0}^{\infty} c_n (x - a)^n$ is an infinite polynomial centred at $a$. It has a radius of convergence $R$, with $0 \le R \le \infty$, such that it converges absolutely for $\lvert x - a \rvert < R$ and diverges for $\lvert x - a \rvert > R$. The ratio test usually finds $R$, and the two endpoints need their own check. Inside the radius you can differentiate and integrate term by term, and the radius stays the same.

Lessons Definition of power series, Pointwise and uniform convergence, Power series theorems

Taylor series

The Taylor series of $f$ at $a$ is the power series whose coefficients match all the derivatives of $f$ at $a$,