Limit, continuity and derivative

2.1Limit of a functionOften examined

Calculus is the mathematics of change. Its first tool is the limit. The limit of $f(x)$ as $x$ approaches a number $a$ is the value that $f(x)$ gets closer and closer to when $x$ gets closer and closer to $a$. We write $f(x)\to L$ as $x\to a$, and read the arrow as "approaches".

A first example

Take $f(x)=\frac{16-x^2}{4+x}$. Its domain is every real number except $-4$, because $x=-4$ makes the denominator zero, so $f(-4)$ is not defined. We can still work out $f(x)$ for values of $x$ near $-4$.

The table suggests that $f(x)$ approaches $8$. Algebra confirms it. For $x\ne -4$ the factor $4+x$ cancels:

Limit (intuitive definition)

If $f(x)$ can be made as close as we like to a finite number $L$ by taking $x$ close enough to $a$ (but not equal to $a$), from both the left and the right, then $L$ is the limit of $f(x)$ as $x\to a$. We write ${\lim }_{x\to a}f(x)=L$.

Left-hand limit

${\lim }_{x\to a^{-}}f(x)$: the value $f(x)$ approaches when $x$ approaches $a$ from the left, that is, through values less than $a$.

Right-hand limit

${\lim }_{x\to a^{+}}f(x)$: the value $f(x)$ approaches when $x$ approaches $a$ from the right, that is, through values greater than $a$.

The limit exists only when both one-sided limits exist and are equal.

Why a limit may fail to exist

• Jump: the left and right limits are different numbers (Example 1, and $\frac{|x|}{x}$ at $0$).
• Unbounded values: $f(x)$ grows without bound near $a$, for example $\frac{1}{x^2}$ near $0$. We may write ${\lim }_{x\to 0}\frac{1}{x^2}=\infty$, but $\infty$ is not a number, so the limit does not exist.
• Oscillation: $f(x)$ keeps jumping between values, for example $\sin \frac{1}{x}$ near $0$ takes every value from $-1$ to $1$ in every small interval around $0$.

2.1Theorems on limits and methods of evaluationOften examined

Tables and graphs show what a limit is, but they are slow. The theorems below let us find limits by algebra. Suppose ${\lim }_{x\to a}f(x)=L_1$ and ${\lim }_{x\to a}g(x)=L_2$ both exist, $c$ is a constant and $n$ is a positive integer.

Direct substitution. Applying these theorems term by term shows that for any polynomial $p$, ${\lim }_{x\to a}p(x)=p(a)$. For a rational function $\frac{p(x)}{q(x)}$ with $q(a)\ne 0$, the limit is $\frac{p(a)}{q(a)}$. So the first thing to try is always to put $x=a$.

A method that always works in the exam

1. Substitute $x=a$. If the answer is a real number (and no denominator is zero), that number is the limit.
2. If you get $\frac{0}{0}$, the form is indeterminate: it tells you nothing yet. Simplify first: factorise and cancel the common factor, or rationalise a surd, or use a standard limit such as $\frac{\sin x}{x}\to 1$. Then substitute again.
3. If you get $\frac{k}{0}$ with $k\ne 0$, the function grows without bound near $a$, so the limit does not exist as a finite number. Check the sign from each side if you need to say $+\infty$ or $-\infty$.

2.1Important limitsOften examined

Some limits have the form \dfrac00 but cannot be simplified by factorising. The two trigonometric ones below are the foundation for the derivatives of \sin x and \cos x (section 2.9). In all of them the angle x is in radians.

Example 2 (book): the two trigonometric limits

Results that follow from the sine limit

Limits that define the number e

The number e\approx2.71828 is the limit of \left(1+\dfrac1n\right)^{n} as n grows. These limits are not in the new book's text, but they are standard and FBISE papers use them (2026 Curriculum 2000 paper, Q.2 (vi) OR).

Proof idea: \left(1+\dfrac ax\right)^{bx}=\left[\left(1+\dfrac ax\right)^{x/a}\right]^{ab} and the bracket tends to e.

Limits at infinity

\lim_{x\to\infty}f(x)=L means f(x) gets close to L as x becomes very large. The basic fact is \lim_{x\to\infty}\dfrac{1}{x^{n}}=0 for n>0. For a rational function, divide the top and bottom by the highest power of x in the denominator.

2.2, 2.2.1Continuity at a numberOften examined

A function is continuous when its graph has no gaps, holes or jumps: you can draw it without lifting the pencil from the paper. The figure shows the five things that can happen at a number, here x=1.

Continuity at a number

A function f is continuous at a number a if all three conditions hold: (i) f(a) is defined; (ii) \lim_{x\to a}f(x) exists; (iii) \lim_{x\to a}f(x)=f(a). If any one condition fails, f is discontinuous at a.

Kinds of discontinuity

Removable discontinuity

\lim_{x\to a}f(x) exists, but f(a) is not defined or f(a)\neq\lim_{x\to a}f(x). Defining (or redefining) f(a) as the limit removes it. Example: \dfrac{x^{2}-1}{x-1} at 1; setting f(1)=2 makes it continuous everywhere.

Jump discontinuity

Both one-sided limits exist but are different, so the graph jumps (Example 1, \dfrac{|x|}{x} at 0). It cannot be removed.

Infinite discontinuity

f(x) grows without bound near a; the graph has a vertical asymptote at x=a, for example \dfrac1{x-3} at 3.

How to test a piecewise function at a joining point

1. Find f(a) from the piece whose condition contains the equality sign at a (\le, \ge or =).
2. Find \lim_{x\to a^-}f(x) using the piece that holds just to the left of a.
3. Find \lim_{x\to a^+}f(x) using the piece that holds just to the right of a.
4. If all three numbers are equal, f is continuous at a. If unknown constants appear, set the three expressions equal and solve.

2.2.2Continuity on an interval

Continuous on an open interval

f is continuous on (a,b) if it is continuous at every number in (a,b).

Continuous on a closed interval

f is continuous on [a,b] if it is continuous on (a,b) and, in addition, \lim_{x\to a^+}f(x)=f(a) (continuous from the right at a) and \lim_{x\to b^-}f(x)=f(b) (continuous from the left at b).

At an endpoint we only look from inside the interval. Half-open intervals such as [a,b) or [a,\infty) are treated the same way: check the one-sided limit at each included endpoint.

Which functions are continuous (key facts)

• If f and g are continuous at a, then cf (c a constant), f+g, f-g, fg and \dfrac fg (when g(a)\neq0) are continuous at a. A composition of continuous functions is continuous.
• Polynomials, \sin x, \cos x and e^{x} are continuous for every real x.
• A rational function is continuous everywhere except where its denominator is zero.
• \tan x and \sec x are discontinuous where \cos x=0, that is, at odd multiples of \dfrac{\pi}{2}; \cot x and \operatorname{cosec}x are discontinuous where \sin x=0, that is, at multiples of \pi.
• \ln x is continuous for x>0; \sqrt{x} is continuous for x\ge0 (from the right at 0).

Intermediate value theorem

The book does not state this theorem, but Exercise 2.2 Q.19 needs it, and it is the reason the root-finding methods of Unit 10 work ("a root lies where the sign changes").

Intermediate value theorem

If f is continuous on [a,b] and N is any number between f(a) and f(b), then there is at least one c in (a,b) with f(c)=N. In particular, if f(a) and f(b) have opposite signs, the equation f(x)=0 has a root in (a,b).

2.3Tangent to a graph and rate of change

To write the equation of the tangent line L to the curve y=f(x) at a point P we need two things: the coordinates of P and the slope m_{\tan} of L. The point is easy: if x=a, then P is (a,f(a)). The slope needs a limit.

Take a second point Q on the curve with x-coordinate a+\Delta x, where \Delta x (read “delta x”) is a small change in x. The line through P and Q is a secant. Its slope is the change in y divided by the change in x:

Tangent line

Let y=f(x) be continuous. The tangent line to the graph at (a,f(a)) is the line through this point with slope m_{\tan}=\lim_{\Delta x\to0}\dfrac{f(a+\Delta x)-f(a)}{\Delta x}=\lim_{\Delta x\to0}\dfrac{\Delta y}{\Delta x}, whenever this limit exists. This slope is also called the slope of the curve at the point.

The four-step method

1. Evaluate f(a) and f(a+\Delta x).
2. Find \Delta y=f(a+\Delta x)-f(a) and simplify.
3. Divide by \Delta x (\Delta x\neq0) and cancel.
4. Let \Delta x\to0: m_{\tan}=\lim_{\Delta x\to0}\dfrac{\Delta y}{\Delta x}.

When a tangent does not exist

• f is discontinuous at x=a (a jump or a hole).
• The graph has a corner at (a,f(a)), where the slopes from the two sides differ.
• The graph has a sharp peak (cusp) at the point.

Rate of change

The slope \dfrac{\Delta y}{\Delta x} of a secant is the average rate of change of f over the interval from a to a+\Delta x. The slope m_{\tan}=\lim_{\Delta x\to0}\dfrac{\Delta y}{\Delta x} is the instantaneous rate of change of f at a. For example, if m_{\tan}=\dfrac1{10} at (a,f(a)), then near a the value of f changes by only about \dfrac1{10} unit for each unit change in x.

2.4Instantaneous velocity

If a bus covers 60 miles in one hour, its average velocity is 60 miles per hour. It did not move at exactly this speed all the time; it slowed down in towns and went faster on open roads. The average does not tell us the velocity at a particular instant.

A runner finishes a 10\,\mathrm{km} race in 1.25\,\mathrm{h}, so v_{\text{ave}}=\dfrac{10}{1.25}=8\,\mathrm{km\,h^{-1}}. In the first half hour she runs 5\,\mathrm{km}: v_{\text{ave}}=\dfrac{5}{0.5}=10\,\mathrm{km\,h^{-1}}. From 0.5\,\mathrm{h} to 0.6\,\mathrm{h} she goes from the 5\,\mathrm{km} mark to the 5.7\,\mathrm{km} mark: v_{\text{ave}}=\dfrac{5.7-5}{0.6-0.5}=7\,\mathrm{km\,h^{-1}}. Shrinking the time interval gives an ever better picture of her velocity at t=0.5\,\mathrm{h}. The limit of this process is the instantaneous velocity.

Instantaneous velocity

Let s=f(t) give the position at time t of an object moving in a straight line. The instantaneous velocity at time t_1 is v(t_1)=\lim_{\Delta t\to0}\dfrac{f(t_1+\Delta t)-f(t_1)}{\Delta t}=\lim_{\Delta t\to0}\dfrac{\Delta s}{\Delta t}, whenever the limit exists. Its SI unit is \mathrm{m\,s^{-1}}.

2.5The derivative and differentiation from first principlesOften examined

The same limit appeared twice: as the slope of a tangent and as an instantaneous velocity. It is so important that it gets its own name and symbol.

Derivative

The derivative of f with respect to x is the function f' given by f'(x)=\lim_{\Delta x\to0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}. Its domain is every x in the domain of f for which the limit exists.

Differentiable

f is differentiable at x if this limit exists (as a finite number).

Differentiation

The process of finding the derivative. Finding it straight from the limit is called differentiation from first principles (or by definition, or the \Delta-method).

At a particular number x_0, f'(x_0) is the slope of the tangent at (x_0,f(x_0)) and the instantaneous rate of change of y with respect to x at x_0.

Method

1. Write f(x) and f(x+\Delta x).
2. Find \Delta y=f(x+\Delta x)-f(x) and simplify so that a factor \Delta x appears.
3. Divide by \Delta x and cancel.
4. Let \Delta x\to0.

2.6, 2.6.1Power, constant and sum rulesOften examined

Differentiating f(x)=5x^{100}+x^{7/5} from first principles would take a long time. The rules below give derivatives quickly. Each is proved once from the definition and then used freely.

The power rule is true for every real n (negative, fractional or irrational), wherever x^{n} is defined. The book states this without proof.

Constant, constant multiple and sum rules

In words: the derivative of a constant is zero (its graph is a horizontal line, slope 0); a constant factor stays outside; and the derivative of a sum or difference is the sum or difference of the derivatives.

Science and business formulas use other letters. The rules are the same; only the names change.

2.7Product and quotient rulesOften examined

The derivative of a product is not the product of the derivatives. With f(x)=x^{2} and g(x)=x^{3}: f'(x)g'(x)=(2x)(3x^{2})=6x^{3}, but f(x)g(x)=x^{5} has derivative 5x^{4}\neq6x^{3}.

2.8Connection between derivative and continuity

• If a function is differentiable at a point, it is automatically continuous at that point.
• The reverse is not always true: a function can be continuous at a point and still not be differentiable there, for example at a sharp corner or a cusp.

2.9Derivatives of trigonometric functionsOften examined

We now find the derivatives of the six trigonometric functions. The angle x is always in radians, because the two limits we need are true only in radians:

2.10Derivatives of inverse trigonometric functionsOften examined

Each inverse trigonometric function is found by the same trick: rewrite y=\sin^{-1}x as x=\sin y, differentiate with respect to x, and express the answer in terms of x. \tan^{-1}x and \cot^{-1}x are differentiable for all real x; the other four are not differentiable at x=\pm1.

2.11Chain rule (derivative of a composite function)Often examined

Consider y=(x^{5}+1)^{2}. Writing it as a product (x^{5}+1)(x^{5}+1) and using the product rule gives

The answer has the pattern: bring down the power 2, keep the bracket with power one less, and multiply by the derivative of the inside, 5x^{4}. This is the power rule for functions.

Power rule for functions

If n is a real number and g is differentiable, then \dfrac{d}{dx}[g(x)]^{n}=n[g(x)]^{n-1}g'(x).

Chain rule

If y=f(u) is a differentiable function of u and u=g(x) is a differentiable function of x, then \dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}=f'(g(x))\,g'(x).

Method: outside, then inside

1. Identify the outer function and the inner function. In \sin(5x) the outer is \sin(\;) and the inner is 5x.
2. Differentiate the outer function, keeping the inside unchanged: \cos(5x).
3. Multiply by the derivative of the inside: \cos(5x)\cdot5.

2.12Implicit differentiationOften examined

Explicit function

The dependent variable is written alone, in terms of the independent variable only: y=f(x). Example: y=\dfrac13x^{3}+\dfrac43.

Implicit function

x and y are mixed in an equation that is not solved for y. Example: 3y-x^{3}-4=0 defines the same y implicitly.

Often we cannot, or do not want to, solve for y (try 5y^{2}+\sin y=x^{2}). Implicit differentiation finds \dfrac{dy}{dx} directly. Take xy=1. Solving first gives y=\dfrac1x and \dfrac{dy}{dx}=-\dfrac1{x^{2}}. Implicitly:

Method

1. Differentiate every term on both sides with respect to x.
2. Whenever you differentiate an expression in y, multiply by \dfrac{dy}{dx} (chain rule): \dfrac{d}{dx}y^{2}=2y\dfrac{dy}{dx}, \dfrac{d}{dx}\sin y=\cos y\dfrac{dy}{dx}. Use the product rule for terms such as xy or x^{3}y^{2}.
3. Collect all \dfrac{dy}{dx} terms on one side and the rest on the other.
4. Factor out \dfrac{dy}{dx} and divide.

2.13, 2.14Derivatives of exponential and logarithmic functionsOften examined

The natural exponential function e^{x} (with e\approx2.71828) has a special property: it is its own derivative. The natural logarithm \ln x=\log_e x is its inverse.

Logarithmic differentiation

When the variable is in both the base and the power, as in x^{x}, neither the power rule nor the exponential rule applies. Take logarithms first, then differentiate implicitly. The same trick simplifies long products and quotients.

2.14Parametric differentiationOften examined

Parametric equations

A curve given by x=f(t) and y=g(t), where both coordinates depend on a third variable t called the parameter (often time or an angle).

2.15DifferentialsOften examined

For a small change \Delta x, the secant slope \dfrac{\Delta y}{\Delta x} is close to the tangent slope f'(x). Multiplying by \Delta x gives a simple estimate of the change in y:

Differential of x

The increment \Delta x of the independent variable is called the differential of x and is written dx, so dx=\Delta x.

Differential of y

For y=f(x), the quantity f'(x)\,dx is called the differential of y and is written dy: dy=f'(x)\,dx.

Geometrically, dy is the rise of the tangent line and \Delta y is the rise of the curve over the same run dx. So \Delta y\approx dy, and the approximation improves as dx gets smaller. Dividing dy=f'(x)\,dx by dx shows that the derivative really is a ratio of two differentials: f'(x)=\dfrac{dy}{dx}.

2.16, 2.22Approximations and errors using differentialsOften examined

If we know f and f' at a convenient number x, differentials predict the value at a nearby number x+\Delta x. Since f(x+\Delta x)=f(x)+\Delta y and \Delta y\approx dy:

This is the tangent-line (linear) approximation. It is the sense in which a derivative lets us approximate a function.

Method

1. Choose the function f and a convenient x near the given number where f(x) is known exactly (a perfect square, 45^{\circ}, and so on).
2. Find dx=\Delta x = given number minus x. For angles, convert dx to radians.
3. Compute f(x) and f'(x).
4. Use f(x+\Delta x)\approx f(x)+f'(x)\,dx.

Absolute, relative and percentage error

When a length is measured with a small possible error dx, any quantity y=f(x) calculated from it inherits an error, approximately dy.

Absolute error

The approximate error in y: dy=f'(x)\,dx (same unit as y).

Relative error

\dfrac{dy}{y}, the error as a fraction of the value (no unit).

Percentage error

\dfrac{dy}{y}\times100\%.

Shortcut: the relative error of a power is the power times the relative error of the measurement. Cube: \dfrac{dV}{V}=3\dfrac{dx}{x}; sphere: \dfrac{dV}{V}=3\dfrac{dr}{r}; square: \dfrac{dA}{A}=2\dfrac{dx}{x}.

2.17, 2.18Higher order derivatives

The derivative f'(x) is itself a function, so we can differentiate it again. The result is the second derivative.

Differentiating again gives the third, fourth and in general the nth derivative:

Increasing and decreasing functionsOften examined

Increasing function

f is increasing on an interval I if for any x_1<x_2 in I, f(x_1)<f(x_2): the graph rises from left to right.

Decreasing function

f is decreasing on I if for any x_1<x_2 in I, f(x_1)>f(x_2): the graph falls from left to right.

The derivative test

• If f'(x)>0 for every x in (a,b), then f is increasing on (a,b).
• If f'(x)<0 for every x in (a,b), then f is decreasing on (a,b).
• If f'(c)=0, the tangent at c is horizontal: c is a stationary point, where f may change from increasing to decreasing or the reverse.

Method for the intervals

1. Find f'(x) and factorise it.
2. Solve f'(x)=0 (and note where f' does not exist). These numbers split the real line into intervals.
3. Pick a test value in each interval and find the sign of f' there (or use the signs of the factors).
4. Write the answer: increasing where f'>0, decreasing where f'<0.

2.19Absolute extremaOften examined

Suppose f is defined on an interval I. The largest and smallest values of f on I (if they exist) are called the extrema of f. There are two kinds: absolute (global) and relative (local).

Absolute maximum

f(c) is the absolute maximum of f if f(x)\le f(c) for every x in the domain of f.

Absolute minimum

f(c) is the absolute minimum of f if f(x)\ge f(c) for every x in the domain of f.

Absolute extrema are also called global extrema. A function may have both, one, or neither.

Extreme value theorem

A function that is continuous on a closed interval [a,b] always has an absolute maximum and an absolute minimum on that interval.

The theorem needs both conditions. On the open interval (1,2), x^{2} gets as close as we like to 1 and 4 but never reaches them. And \dfrac1x on [-1,1] has no extrema because it is not continuous at 0.

Closed interval method for absolute extrema

1. Check that f is continuous on [a,b].
2. Find the critical values of f that lie inside (a,b) (section 2.19.1).
3. Evaluate f at each critical value and at the endpoints a and b.
4. The largest of these values is the absolute maximum; the smallest is the absolute minimum.

2.19.1Relative extrema and critical valuesOften examined

A function with no absolute extrema can still have hills and valleys. Near the top of a hill, f(c) is the largest value in a small neighbourhood of c, even if the function goes higher far away.

Relative maximum

f(c) is a relative (local) maximum of f if f(x)\le f(c) for every x in some open interval that contains c.

Relative minimum

f(c) is a relative (local) minimum of f if f(x)\ge f(c) for every x in some open interval that contains c.

Result. If f has a relative extremum at c, then either f'(c)=0 or f'(c) does not exist. At a smooth hill or valley the tangent is horizontal; at a sharp corner there is no tangent.

Critical value

A number c in the domain of f for which f'(c)=0 or f'(c) does not exist. The point (c,f(c)) is a critical point. When f'(c)=0 it is also called a stationary point.

First derivative test

This test is not in the book, but it is the natural way to classify a critical value, and it works even when f'(c) does not exist.

• If f' changes from positive to negative as x increases through c, then f(c) is a relative maximum.
• If f' changes from negative to positive, then f(c) is a relative minimum.
• If f' does not change sign, f(c) is not an extremum.

2.20, 2.21Concavity, points of inflection and the second derivative testOften examined

A curve that bends upward like a cup is concave upward; it “holds water”. A curve that bends downward like a cap is concave downward; it “spills water”. On a concave upward arc the slope f' increases, so f''>0; on a concave downward arc the slope decreases, so f''<0.

Test for concavity

Let f'' exist on (a,b). If f''(x)>0 for all x in (a,b), the graph of f is concave upward on (a,b). If f''(x)<0 for all x in (a,b), it is concave downward on (a,b).

Point of inflection

Let f be continuous at c. The point (c,f(c)) is a point of inflection if there is an open interval (a,b) containing c such that the graph is concave upward on (a,c) and concave downward on (c,b), or the reverse. In short, the concavity changes at c.

Second derivative test

Let f'(c)=0 and let f'' exist on an interval containing c. (i) If f''(c)>0, then f(c) is a relative minimum. (ii) If f''(c)<0, then f(c) is a relative maximum. (iii) If f''(c)=0, the test fails; use the first derivative test.

Exam template: critical points, classification and absolute extrema

1. Write f'(x), factorise it, and solve f'(x)=0 inside the given interval (for trigonometric functions list every solution in the interval).
2. Write f''(x) and find its sign at each critical value; state maximum or minimum.
3. Find f at each critical value and at the endpoints.
4. State the absolute maximum and minimum with the x values where they occur.

2.22Applications of derivatives

Many real quantities change: the speed of a rocket, inflation of prices, the number of bacteria in a culture, the voltage of a signal. Derivatives measure how fast they change, and they find the largest or smallest value of a quantity (optimisation). Error estimates with differentials (Example 49 of the book) are covered in the section on approximations and errors.

Related rates

When two quantities are linked by an equation and both change with time t, differentiating the equation with respect to t links their rates.

1. Draw a figure and name the variables.
2. Write the equation that links them.
3. Differentiate both sides with respect to t (chain rule).
4. Substitute the known values and rates only after differentiating.
5. Answer with units.

Optimisation

1. Name the variables and write the quantity to be made largest or smallest.
2. Use the given condition (constraint) to write it as a function of one variable, and state the allowed interval.
3. Find the critical values.
4. Test them (second derivative test, or compare with the endpoint values).
5. Answer the question that was asked, with units.

Price growth and inflation

With a continuous annual inflation rate r, the price level at time t years is P(t)=P_0e^{rt}, where P_0 is the initial price. The derivative gives how fast prices are rising at time t:

Inflation over a period. The percentage rise in price from time t_1 to t_2 is \dfrac{P(t_2)-P(t_1)}{P(t_1)}\times100\%. For the model P_0e^{rt} this is \left(e^{r(t_2-t_1)}-1\right)\times100\%; with r=0.03 over 5 years it is (e^{0.15}-1)\times100\%\approx16.2\%. For actual prices, if a basket of goods cost Rs 2000 in 2020 and Rs 2600 in 2025, inflation over the period is \dfrac{600}{2000}\times100\%=30\%.

Straight-line models: marginal cost and revenue

When a quantity changes at a constant rate its graph is a straight line, and the derivative is the constant slope. In economics, the marginal revenue R'(x) and marginal cost C'(x) are the extra revenue and extra cost from one more unit.

Straight-line depreciation

A machine loses value as it ages. In the straight-line method it loses the same amount D every year: D=\dfrac{C-S}{n}, where C is the cost, S the salvage (scrap) value and n the useful life in years. Its book value after t years is V(t)=C-Dt, and \dfrac{dV}{dt}=-D: the value falls at a constant rate.

Formula sheet

Key terms

Limit

The number L that f(x) approaches as x approaches a from both sides, written \lim_{x\to a}f(x)=L.

Left-hand limit

The value f(x) approaches as x\to a through values less than a, written \lim_{x\to a^-}f(x).

Right-hand limit

The value f(x) approaches as x\to a through values greater than a, written \lim_{x\to a^+}f(x).

Indeterminate form

An expression such as \dfrac00 obtained by substitution; it gives no information until the function is simplified.

Continuous at a number

f is continuous at a if f(a) is defined, \lim_{x\to a}f(x) exists and \lim_{x\to a}f(x)=f(a).

Discontinuous

Not continuous: at least one of the three conditions of continuity fails at the number.

Continuous on an interval

Continuous at every number of an open interval; on a closed interval [a,b] also continuous from the right at a and from the left at b.

Removable discontinuity

A discontinuity where the limit exists; defining f(a) equal to the limit removes it.

Jump discontinuity

A discontinuity where the left-hand and right-hand limits exist but are different.

Continuous extension

The function obtained by defining (or redefining) f(a) as \lim_{x\to a}f(x) at a removable discontinuity.

Intermediate value theorem

A function continuous on [a,b] takes every value between f(a) and f(b); so a sign change means a root.

Secant line

A line through two points of a curve; its slope \dfrac{\Delta y}{\Delta x} is the average rate of change.

Tangent line

The line through (a,f(a)) with slope \lim_{\Delta x\to0}\dfrac{\Delta y}{\Delta x}, whenever this limit exists.

Average rate of change

\dfrac{f(a+\Delta x)-f(a)}{\Delta x}, the change in f per unit change in x over an interval.

Instantaneous rate of change

The limit of the average rate of change as \Delta x\to0; the derivative at a point.

Instantaneous velocity

v(t)=\lim_{\Delta t\to0}\dfrac{\Delta s}{\Delta t}, the rate of change of position at an instant.

Derivative

f'(x)=\lim_{\Delta x\to0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}, the slope of the curve and the rate of change of f.

Differentiable

A function is differentiable at x if its derivative exists (as a finite limit) there.

Differentiation

The process of finding a derivative; “from first principles” means directly from the limit definition.

Chain rule

\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}: the derivative of a composite function.

Explicit function

A function written as y=f(x), with y alone on one side.

Implicit function

A relation in which x and y are mixed, such as 5y^{2}+\sin y=x^{2}.

Implicit differentiation

Differentiating both sides of an equation in x and y with respect to x, treating y as a function of x.

Parametric equations

Equations x=f(t), y=g(t) that give both coordinates in terms of a parameter t.

Logarithmic differentiation

Taking logarithms before differentiating, used for functions such as x^{x}.

Differential

dx=\Delta x is the differential of x; dy=f'(x)\,dx is the differential of y.

Absolute error

The approximate error dy in a calculated quantity caused by an error dx in a measurement.

Relative error

The error as a fraction of the value, \dfrac{dy}{y}.

Percentage error

The relative error times 100, \dfrac{dy}{y}\times100\%.

Higher order derivative

A derivative of a derivative: f'', f''', f^{(4)}, and so on.

Increasing function

f is increasing on I if x_1<x_2 in I implies f(x_1)<f(x_2).

Decreasing function

f is decreasing on I if x_1<x_2 in I implies f(x_1)>f(x_2).

Absolute maximum

f(c) with f(x)\le f(c) for every x in the domain; also called the global maximum.

Absolute minimum

f(c) with f(x)\ge f(c) for every x in the domain; also called the global minimum.

Extreme value theorem

A function continuous on a closed interval [a,b] has an absolute maximum and an absolute minimum on it.

Relative maximum

f(c) with f(x)\le f(c) for every x in some open interval containing c (local maximum).

Relative minimum

f(c) with f(x)\ge f(c) for every x in some open interval containing c (local minimum).

Critical value

A number c in the domain of f where f'(c)=0 or f'(c) does not exist; (c,f(c)) is a critical point.

Stationary point

A point where f'(c)=0, so the tangent is horizontal.

Concave upward

The graph bends up like a cup (“holds water”); f''>0.

Concave downward

The graph bends down like a cap (“spills water”); f''<0.

Point of inflection

A point on a continuous graph where the concavity changes from upward to downward or the reverse.

Optimisation

Finding the largest or smallest value of a quantity, usually by locating critical values.

Marginal cost

C'(x), the approximate cost of producing one more unit.

Marginal revenue

R'(x), the approximate revenue from selling one more unit.

Break-even point

The number of units at which revenue equals cost, R(x)=C(x).

Inflation rate

The rate at which the price level rises; with P=P_0e^{rt}, r is the continuous annual rate.

Straight-line depreciation

A method in which an asset loses the same value D=\dfrac{C-S}{n} every year.