UBC MATH 100, topic by topic

Say which of two powers of $x$ is larger, for $0 < x < 1$ and for $x > 1$, and sketch $y = x^{n}$ for any real $n$

UBC Math 100 §1.1 · Practise it: 5 questions

Pick out the term of a polynomial that dominates near zero, and the one that dominates for large x

UBC Math 100 §1.1 · Practise it: 8 questions

Sketch a two-term polynomial, such as $x^{2} - 3x^{4}$, from its dominant terms

UBC Math 100 §1.2 · Practise it: 5 questions

Use the order of growth (a logarithm below any positive power, any power below $e^{x}$) to sketch a function such as $e^{x} - x$

UBC Math 100 §1.5 · Practise it: 3 questions

Sketch the familiar functions from memory: $e^{x}$, $\log x$, $\sin x$, $\cos x$, $\tan x$, $1/x$, $\sqrt{x}$ and $|x|$

UBC Math 100 §1.5 · Practise it: 3 questions

Find where a formula is defined, and compose two functions

Assumed from precalculus · UBC Math 100 §1.5 · Practise it: 5 questions

Predict what a shift, stretch or reflection does to a graph

Assumed from precalculus · UBC Math 100 §1.5 · Practise it: 1 question

Solve the exponential, logarithmic and trigonometric equations later topics take for granted

Assumed from precalculus · UBC Math 100 §1.5 · Practise it: 8 questions

Say what an inverse function is, when one exists, and find it

UBC Math 100 §4.6 · Practise it: 5 questions

Explain in words and with a picture what a limit, a one-sided limit and an infinite limit mean

UBC Math 100 §2.1 · Practise it: 5 questions

Read limits off a graph, and say when a limit does not exist

UBC Math 100 §2.1 · Practise it: 4 questions

Work out a limit from each side, and say the limit exists only when the two agree

UBC Math 100 §2.1 · Practise it: 6 questions

Decide when substituting the point is enough

UBC Math 100 §2.1.1 · Practise it: 4 questions

Evaluate limits of polynomial, rational, trigonometric, exponential and logarithmic functions, resolving a $0/0$ form when substitution gives one

UBC Math 100 §2.1.1 · Practise it: 10 questions

Trap a bounded, oscillating factor between two functions with the same limit (the squeeze theorem)

Beyond the course text · CLP-1 §1.4 · Practise it: 1 question

Find limits as $x \to \infty$ and $x \to -\infty$, square roots included

UBC Math 100 §2.1.2 · Practise it: 11 questions

Find the vertical and horizontal asymptotes of a function, by limits or by dominant terms, and say what each means

UBC Math 100 §2.2 · Practise it: 9 questions

Explain why a graph can cross its horizontal asymptote

UBC Math 100 §2.2 · Practise it: 1 question

State what it means for a function to be continuous at a point and on its domain, in words and with limits

UBC Math 100 §2.3 · Practise it: 6 questions

Classify a discontinuity as removable, jump or infinite

UBC Math 100 §2.3 · Practise it: 7 questions

Say where a function is continuous, and justify it

UBC Math 100 §2.3 · Practise it: 5 questions

Choose the constants that make a piecewise function continuous

UBC Math 100 §2.3 · Practise it: 11 questions

Use the Intermediate Value Theorem to show an equation has a solution in an interval

Beyond the course text · CLP-1 §1.6 · Practise it: 4 questions

Find a line from two points, from a point and a slope, or from a graph, and read the slopes of a piecewise-linear function

UBC Math 100 §3.1 · Practise it: 5 questions

Compute the average rate of change of a function over an interval

UBC Math 100 §3.2 · Practise it: 2 questions

Explain the derivative as the limit of difference quotients, and use that limit to find a derivative and a tangent line

UBC Math 100 §3.3 · Practise it: 14 questions

Sketch the graph of $f'$ from the graph of $f$

UBC Math 100 §3.3 · Practise it: 2 questions

Read a derivative as an instantaneous rate of change, with its units

UBC Math 100 §3.3 · Practise it: 4 questions

Find where a function fails to be differentiable: a corner, a cusp, a vertical tangent or a break

UBC Math 100 §3.3 · Practise it: 8 questions

Split a derivative over sums and constant multiples, and show from the definition why that is allowed

UBC Math 100 §4.1 · Practise it: 3 questions

State and use the power rule for any exponent, and prove it for whole-number powers from the definition

UBC Math 100 §4.1 · Practise it: 6 questions

Differentiate a product without dropping a term

UBC Math 100 §4.1 · Practise it: 4 questions

Differentiate a quotient, the right way round

UBC Math 100 §4.1 · Practise it: 7 questions

Differentiate first and substitute the point second

UBC Math 100 §4.1 · Practise it: 7 questions

Find a tangent line from a derivative, and run the idea backwards to find where a tangent has a given property

UBC Math 100 §3.3.3 · Practise it: 5 questions

Recognise a limit as a derivative in disguise

UBC Math 100 §3.3 · Practise it: 1 question

Show a claim about derivatives is false with a counterexample, and say why an example cannot prove one true

UBC Math 100 §4.1 · Practise it: 1 question

Compute second and higher derivatives

UBC Math 100 §3.4 · Practise it: 3 questions

Show from the definition that the derivative of $a^{x}$ is a constant times $a^{x}$, and describe $e^{x}$ as the exponential that is its own derivative

UBC Math 100 §3.5 · Practise it: 8 questions

Differentiate sine, cosine and tangent, and get secant, cosecant and cotangent from them

UBC Math 100 §4.2 · Practise it: 12 questions

Explain where the trigonometric derivatives come from: the limit of $\sin x / x$ and the addition formulas

UBC Math 100 §4.2 · Practise it: 4 questions

Differentiate a composition with the chain rule, nested ones included

UBC Math 100 §4.3 · Practise it: 15 questions

Differentiate logarithms

UBC Math 100 §4.4 · Practise it: 6 questions

Decide when logarithmic differentiation makes a derivative easier (a long product or quotient, a variable power), and carry it out

UBC Math 100 §4.4 · Practise it: 11 questions

Explain implicit differentiation as the chain rule with $y$ a function of $x$

UBC Math 100 §4.5 · Practise it: 4 questions

Find the slope and the tangent line of a curve defined implicitly

UBC Math 100 §4.5 · Practise it: 14 questions

Find the derivative of an inverse function at a point from the derivative of the function itself

UBC Math 100 §4.6 · Practise it: 1 question

Evaluate $\arcsin$, $\arccos$ and $\arctan$ at standard values, and sketch $y = \arctan x$

UBC Math 100 §4.7 · Practise it: 4 questions

Derive and use the derivatives of arcsine, arccosine and arctangent

UBC Math 100 §4.7 · Practise it: 11 questions

Name each changing quantity, and say which rate is given and which is wanted

UBC Math 100 §5 · Practise it: 14 questions

Write a relationship between the quantities that holds at every instant

UBC Math 100 §5 · Practise it: 17 questions

Use the fixed geometry to remove a variable before differentiating

UBC Math 100 §5 · Practise it: 5 questions

Differentiate the relationship with respect to time

UBC Math 100 §5 · Practise it: 24 questions

Substitute the instant only at the end, and say what the sign of the rate means

UBC Math 100 §5 · Practise it: 15 questions

Recognise the two forms the rule applies to directly, $0/0$ and $\infty/\infty$, and refuse it anywhere else

UBC Math 100 §6.1 · Practise it: 7 questions

Evaluate a limit with the rule, one-sided and at infinity included, applying it again while the form stays indeterminate

UBC Math 100 §6.2 · Practise it: 13 questions

Rewrite a $0 \cdot \infty$ product as a quotient before using the rule

UBC Math 100 §6.2 · Practise it: 3 questions

Compare the rule with dominant-term reasoning, and use whichever is shorter

UBC Math 100 §6 · Practise it: 4 questions

Resolve the forms $\infty - \infty$, $1^{\infty}$, $0^{0}$ and $\infty^{0}$ by a common denominator or a logarithm

Optional in the course text · UBC Math 100 §6.3 · Practise it: 5 questions

Start a sketch from the domain, intercepts and asymptotes

UBC Math 100 §7.1 · Practise it: 5 questions

Find the sign of a factored function quickly, from where each factor changes sign

UBC Math 100 §7.1 · Practise it: 6 questions

Find where a function increases and decreases from the sign of $f'$

UBC Math 100 §7.2 · Practise it: 12 questions

Say what concavity means, and find where a function is concave up or down, and its inflection points, from $f''$

UBC Math 100 §7.3 · Practise it: 13 questions

Recognise an even, odd or periodic function, and use the symmetry in a sketch

UBC Math 100 §7.4 · Practise it: 5 questions

Draw a complete sketch, unaided, from $f$, $f'$ and $f''$

UBC Math 100 §7.5 · Practise it: 3 questions

Find a slant asymptote by dividing out the polynomial part

Beyond the course text · Practise it: 2 questions

Find the critical and singular points of a function, and identify its local maxima and minima

UBC Math 100 §8.1 · Practise it: 4 questions

Find the global maximum and minimum on a closed interval, checking the endpoints

UBC Math 100 §8.2 · Practise it: 4 questions

Separate the quantity being optimised from the constraint

UBC Math 100 §8.3 · Practise it: 9 questions

Use the constraint to write the objective in one variable

UBC Math 100 §8.3 · Practise it: 8 questions

Find the domain the situation allows

UBC Math 100 §8.3 · Practise it: 6 questions

Justify that a critical point really is the extreme value: finding one is not enough

UBC Math 100 §8.2 · Practise it: 11 questions

Answer the question asked, and interpret the optimum in its context

UBC Math 100 §8.4 · Practise it: 7 questions

Estimate a value that is hard to compute with a tangent line, choosing a centre where everything is easy

UBC Math 100 §9.2 · Practise it: 11 questions

Approximate an irrational number such as $\sqrt{10}$ by a rational one, choosing both the function and the centre

UBC Math 100 §9.2 · Practise it: 3 questions

Estimate a change with $\Delta y \approx f'(a)\,\Delta x$

UBC Math 100 §9.2 · Practise it: 2 questions

Say whether a tangent-line estimate runs high or low, from the concavity between the centre and the point

UBC Math 100 §9.2 · Practise it: 2 questions

Say what a degree $n$ approximation of a function is, and build one at any centre

UBC Math 100 §9.4 · Practise it: 8 questions

State the Maclaurin polynomials of $1/(1 - x)$, $e^{x}$, $\cos x$, $\sin x$ and $\log(1 + x)$

UBC Math 100 §9.5 · Practise it: 6 questions

Get a new Maclaurin polynomial from a known one by substituting, multiplying or differentiating, rather than from scratch

UBC Math 100 §9.5.1 · Practise it: 6 questions

Evaluate a limit by replacing each function with its Taylor polynomial

UBC Math 100 §9.5.2 · Practise it: 1 question

Read a derivative at the centre back off a coefficient

UBC Math 100 §9.4 · Practise it: 3 questions

Bound the error of a Taylor approximation, find where it meets a tolerance, or find the degree that does

Flavour A only · UBC Math 100 §9.6 · Practise it: 3 questions

Find a whole number close to a root to start from

UBC Math 100 §10.1 · Practise it: 1 question

Find where the tangent line at a point crosses the x-axis, and explain how repeating that finds a root

UBC Math 100 §10.1 · Practise it: 4 questions

Write down the iteration, say what each term is, and carry out a step or two by hand

UBC Math 100 §10.1 · Practise it: 9 questions

Recognise a starting point from which the method stalls, cycles or runs away

UBC Math 100 §10.1 · Practise it: 5 questions

Run the iteration in a spreadsheet

Done in a spreadsheet, not practised here · UBC Math 100 §10.2

Model growth and decay with $y' = ky$, finding $k$ from one measurement, and work with half-lives and doubling times

UBC Math 100 §11.3 · Practise it: 6 questions

Solve $y' = a - by$, by its formula or a change of variables, and interpret the solution: cooling, inflow and outflow

UBC Math 100 §11.4 · Practise it: 5 questions

Use an initial condition to pick out the one solution that fits

UBC Math 100 §11.4 · Practise it: 5 questions

Say how a differential equation differs from an algebraic one, and check whether a function solves one

UBC Math 100 §11.2 · Practise it: 7 questions

Find the slope of the solution through a given point, and draw a short piece of it

UBC Math 100 §12.2 · Practise it: 4 questions

Read a slope field and sketch solutions on it

UBC Math 100 §12.2 · Practise it: 4 questions

Explain Euler’s method and each term in its formula, and take a few steps by hand

UBC Math 100 §12.3 · Practise it: 5 questions

Compare an Euler estimate with the exact solution, and say whether it runs high or low

UBC Math 100 §12.3 · Practise it: 2 questions

Run Euler’s method in a spreadsheet

Done in a spreadsheet, not practised here · UBC Math 100 §12.4

Say what a steady state is, and find the steady states of a simple differential equation

UBC Math 100 §13.1 · Practise it: 8 questions

Draw a state-space diagram and describe from it what every solution does

UBC Math 100 §13.1 · Practise it: 5 questions

Decide whether a steady state is stable, from the diagram or from the sign of the slope

UBC Math 100 §13.1 · Practise it: 5 questions

Model a population with the logistic equation, and read off its carrying capacity and long-run behaviour

UBC Math 100 §13.2 · Practise it: 6 questions

Place points in three dimensions, and recognise the planes where x, y or z is constant

UBC Math 100 §14.1 · Practise it: 2 questions

Evaluate a function of two variables $z = f(x, y)$ at given points

UBC Math 100 §14.2 · Practise it: 1 question

Use traces and level curves to understand a surface, and match a function to its graph

UBC Math 100 §14.3 · Practise it: 3 questions

Compute partial derivatives, and read one as the steepness of the surface in one direction

UBC Math 100 §15.1 · Practise it: 12 questions

Compute second partial derivatives, and know the mixed ones agree for well-behaved functions

UBC Math 100 §15.2 · Practise it: 2 questions

Find the critical points of a function of two variables, and picture them as flat spots

UBC Math 100 §16.1 · Practise it: 5 questions

Classify critical points as maxima, minima or saddle points with the second derivative test, and say what a saddle point is

UBC Math 100 §16.1 · Practise it: 4 questions

Find the absolute extremes on a closed region: critical points inside, and the boundary reduced to one variable

UBC Math 100 §16.2 · Practise it: 1 question

Find the extremes along a constraint with Lagrange multipliers

UBC Math 100 §16.3 · Practise it: 6 questions

Choose between Lagrange multipliers and substituting the constraint, and say when a constraint has no extreme

UBC Math 100 §16.3 · Practise it: 2 questions