A limit describes what a function is heading toward, even if it never gets there. It is the idea that the whole of calculus is built on.
➡️ What a limit says
The notation limx→a f(x) = L means f(x) gets as close as we like to L when x gets close to a (from both sides). The value of f at a itself does not matter.
Direct substitution first. If it gives 0/0, factor or simplify, then substitute again.
🖼️ See it
f(1) is undefined, but as x gets close to 1 from either side f(x) gets close to 2, so the limit exists. The graph is the line y = x + 1 with a hole.
🧵 Continuity
A function is continuous at a if f(a) exists, the limit as x→a exists, and the two are equal. On a graph you could draw it without lifting your pen.
f(x) = x + 1 for x < 2, kx for x ≥ 2
left limit at 2: 2 + 1 = 3
right value: 2k
continuous when 2k = 3 → k = 1.5
Secant slope of x² from 1 to 3:
(9 − 1)/(3 − 1) = 4 → as the gap shrinks it becomes the derivative
✏️ Quick check · 1 of 7
Jump
A function jumps from 1 to 3 at x = 2. At x = 2 the limit:
Compare the two sides.
✏️ Quick check · 2 of 7
Continuity condition
Which is required for f to be continuous at x = a?
No jump, no hole, no break.
✍️ Type your answer · 3 of 7
Evaluate lim (x² − 4)/(x − 2) as x approaches 2.
What is the limit?
Enter an exact answer, without rounding.
Factor and cancel.
✍️ Type your answer · 4 of 7
Evaluate lim (x² − 1)/(x − 1) as x approaches 1.
What is the limit?
Enter an exact answer, without rounding.
Difference of squares.
✍️ Type your answer · 5 of 7
Evaluate lim (3x + 1)/(x − 5) as x approaches infinity.
What is the limit?
Enter an exact answer, without rounding.
Compare the leading coefficients.
✍️ Type your answer · 6 of 7
Evaluate lim (√x − 2)/(x − 4) as x approaches 4.
What is the limit, as a decimal?
Enter an exact answer, without rounding.
Factor x − 4 as a difference of squares.
✍️ Type your answer · 7 of 7
f(x) = x + 1 for x < 2 and f(x) = kx for x ≥ 2.
For which k is f continuous at x = 2?
Enter an exact answer, without rounding.
Set the two pieces equal at x = 2.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
lim (5/x) as x tends to infinity equals:
Question 2 of 5
A removable discontinuity is a:
Question 3 of 5
The limit as x→a of f(x) depends on:
Question 4 of 5
lim (x² + 2x) as x tends to 3 equals:
Question 5 of 5
The slope of a secant line becomes the slope of the tangent when:
Lesson 2 · Calculus
📈 Differentiation Rules
The derivative f′(x) gives the slope of a curve at each point. A small set of rules lets you differentiate almost any formula quickly.
🧰 The basic rules
The derivative is the limit of the secant slope: f′(x) = lim (f(x + h) − f(x)) / h. In practice we use rules.
Take derivatives term by term. For a product use the product rule; for a function of a function multiply by the derivative of the inside.
🔗 Product, quotient and chain
Product: (uv)′ = u′v + uv′
Quotient: (u/v)′ = (u′v − uv′) / v²
Chain: [f(g(x))]′ = f′(g(x)) · g′(x)
x²(x + 1): 2x(x + 1) + x²·1 = 3x² + 2x at x = 2: 16
x/(x + 1): (1·(x + 1) − x·1)/(x + 1)² = 1/(x + 1)² at x = 1: 0.25
(2x + 1)³: 3(2x + 1)²·2 = 6(2x + 1)² at x = 1: 54
e³ˣ: 3e³ˣ at x = 0: 3
Chain rule = outside × inside. Differentiate the outer function, keep the inside, then multiply by the derivative of the inside.
✏️ Quick check · 1 of 7
Cosine derivative
The derivative of cos x is:
Not the same as the sine rule.
✏️ Quick check · 2 of 7
Which rule
To differentiate (3x + 2)⁵, the best rule is the:
There is an inside function 3x + 2.
✍️ Type your answer · 3 of 7
f(x) = 3x⁴ − 2x
What is f′(2)?
Enter an exact answer, without rounding.
Use the power rule term by term.
✍️ Type your answer · 4 of 7
f(x) = x²(x + 1). Use the product rule.
What is f′(2)?
Enter an exact answer, without rounding.
u = x², v = x + 1.
✍️ Type your answer · 5 of 7
f(x) = x/(x + 1). Use the quotient rule.
What is f′(1), as a decimal?
Enter an exact answer, without rounding.
Numerator: (x + 1) − x = 1.
✍️ Type your answer · 6 of 7
f(x) = (2x + 1)³. Use the chain rule.
What is f′(1)?
Enter an exact answer, without rounding.
Do not forget the inside derivative 2.
✍️ Type your answer · 7 of 7
f(x) = sin(2x)
What is f′(0)?
Enter an exact answer, without rounding.
Chain rule with inside 2x.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
The derivative of x⁵ is:
Question 2 of 5
The derivative of eˣ is:
Question 3 of 5
The derivative of ln x is:
Question 4 of 5
The derivative of a constant is:
Question 5 of 5
The derivative of e²ˣ is:
Lesson 3 · Calculus
⛰️ Applications of Derivatives
Derivatives find tangent lines, turning points and the best possible value of a quantity. This is where calculus starts solving real problems.
📍 Tangents and turning points
The slope of the tangent at x = a is f′(a). A stationary point is where f′(x) = 0. The second derivative tells the type: f″ > 0 is a minimum, f″ < 0 is a maximum.
y = x³ at x = 1
slope = 3x² = 3, point (1, 1)
tangent: y − 1 = 3(x − 1) → y = 3x − 2
at x = 2: y = 4
f(x) = x³ − 12x
f′ = 3x² − 12 = 0 → x = ±2
f″ = 6x: at x = 2 it is 12 > 0, a minimum
🖼️ See it
Where the gradient is zero the curve turns: a maximum on the left, a minimum on the right.
🏆 Optimisation
To find a maximum or minimum: (1) write the quantity in one variable, (2) set the derivative to 0, (3) check it is the type you want.
A farmer has 60 m of fence for three sides of a field
(a barn forms the fourth side).
width x, length 60 − 2x
A = x(60 − 2x) = 60x − 2x²
A′ = 60 − 4x = 0 → x = 15
A = 15 · 30 = 450 m²
Motion: s = t³ − 6t² + 9t
v = s′ = 3t² − 12t + 9
a = v′ = 6t − 12 at t = 4: 12
Rates of change: velocity is the derivative of position and acceleration is the derivative of velocity.
✏️ Quick check · 1 of 7
Second derivative test
At a stationary point, f″(x) > 0 means the point is a:
Think of a smile shape.
✏️ Quick check · 2 of 7
Stationary point
A stationary point is where:
Slope zero.
✍️ Type your answer · 3 of 7
y = x² at the point where x = 3.
What is the slope of the tangent?
Enter an exact answer, without rounding.
Differentiate and substitute.
✍️ Type your answer · 4 of 7
The tangent to y = x³ at x = 1 is y = 3x − 2.
What is the tangent’s y-value at x = 2?
Enter an exact answer, without rounding.
Substitute x = 2 into the line.
✍️ Type your answer · 5 of 7
f(x) = x² − 6x + 5. Solve f′(x) = 0.
At what x does the minimum occur?
Enter an exact answer, without rounding.
Set the derivative to zero.
✍️ Type your answer · 6 of 7
f(x) = x³ − 12x has stationary points where 3x² − 12 = 0.
At which positive x does it have a minimum?
Enter an exact answer, without rounding.
Use the second derivative.
✍️ Type your answer · 7 of 7
A rectangle has a perimeter of 40 cm, so its sides are x and 20 − x.
What is the largest possible area, in cm²?
Enter an exact answer, without rounding.
cm²
The best rectangle is a square.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
The derivative of position with respect to time is:
Question 2 of 5
For a maximum, the sign of f″ at the stationary point is:
Question 3 of 5
The first step in an optimisation problem is usually to:
Question 4 of 5
A tangent line to a curve at a point has slope equal to:
Question 5 of 5
If s = t³ − 6t² + 9t, the acceleration is:
Lesson 4 · Calculus
↩️ Integration
Integration reverses differentiation. Given a rate of change, it recovers the quantity itself, and it is also the tool for accumulating totals.
🔄 Antiderivatives
An antiderivative of f is a function F with F′ = f. Every antiderivative differs by a constant, so we write + C.
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1)
∫ 1/x dx = ln|x| + C
∫ eᵏˣ dx = eᵏˣ/k + C
∫ cos x dx = sin x + C
∫ sin x dx = −cos x + C
∫(3x² + 2) dx = x³ + 2x + C
if F(0) = 1, then C = 1, so F(2) = 8 + 4 + 1 = 13
🖼️ See it
The integral of x² is x³/3. Evaluating between 0 and 3 gives 27/3 − 0 = 9, the shaded area.
🎒 Substitution and initial values
For a composite, substitution turns it into a basic integral: let u be the inside function and make sure du appears.
∫ 2x (x² + 1)³ dx u = x² + 1, du = 2x dx
= ∫ u³ du = u⁴/4 + C = (x² + 1)⁴/4 + C
f′(x) = 4x − 3 and f(1) = 2
f(x) = 2x² − 3x + C
2 − 3 + C = 2 → C = 3
f(2) = 8 − 6 + 3 = 5
Check by differentiating. If you differentiate your answer you must get back the integrand.
✏️ Quick check · 1 of 7
Sine integral
∫ cos x dx equals:
Reverse the derivative.
✏️ Quick check · 2 of 7
Exponential integral
∫ e²ˣ dx equals:
Reverse the chain rule.
✏️ Quick check · 3 of 7
Why +C
Why do we add + C to an indefinite integral?
Think about d/dx of a constant.
✍️ Type your answer · 4 of 7
∫ x⁴ dx = a·x⁵ + C.
What is a, as a decimal?
Enter an exact answer, without rounding.
Divide by the new power.
✍️ Type your answer · 5 of 7
F′(x) = 3x² + 2 and F(0) = 1.
What is F(2)?
Enter an exact answer, without rounding.
Use F(0) to find C.
✍️ Type your answer · 6 of 7
f′(x) = 4x − 3 and f(1) = 2.
What is f(2)?
Enter an exact answer, without rounding.
Integrate, then find C.
✍️ Type your answer · 7 of 7
∫ 2x(x² + 1)³ dx = (x² + 1)⁴/4 + C.
With C = 0, what is the value at x = 1?
Enter an exact answer, without rounding.
Substitute x = 1.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
∫ 1/x dx equals:
Question 2 of 5
If F′(x) = f(x), then F is called:
Question 3 of 5
∫ 6t dt gives:
Question 4 of 5
Integration by substitution is the reverse of the:
Question 5 of 5
If velocity is v(t), position is found by:
Lesson 5 · Calculus
📐 Definite Integrals & Area
A definite integral adds up infinitely many thin slices between two limits. It gives area, total distance, average values and volumes.
📏 The fundamental theorem
∫ab f(x) dx = F(b) − F(a), where F is any antiderivative. If f ≥ 0 this is the area under the curve between a and b.
∫₁³ 2x dx = [x²]₁³ = 9 − 1 = 8
∫₀³ x² dx = [x³/3]₀³ = 9
∫₀^π sin x dx = [−cos x]₀^π = 1 + 1 = 2
Average value of f on [a, b] = (1/(b − a)) ∫ f
x² on [0, 3]: (1/3)·9 = 3
🖼️ See it
∫ from 0 to 2 of x² dx = [x³/3] = 8/3 ≈ 2.67 square units.
🧮 Area between curves, volume and trapezium rule
Area between two curves: integrate top − bottom. Rotating a curve about the x-axis gives volume V = π ∫ y² dx. When no antiderivative is easy, approximate with the trapezium rule.
between y = 2x and y = x on [0, 4]:
∫(2x − x) dx = [x²/2]₀⁴ = 8
y = x rotated about the x-axis, 0 to 3:
V/π = ∫ x² dx = 9
Trapezium rule, h = 1, y = 2, 4, 6:
(h/2)(y₀ + 2y₁ + y₂) = 0.5(2 + 8 + 6) = 8
v = 2t + 1 over t = 0..4: distance = [t² + t] = 20
Signed area: a region below the x-axis counts as negative, so split the integral at each root if you want the total area.
✏️ Quick check · 1 of 7
Fundamental theorem
The definite integral of f from a to b equals:
Upper minus lower.
✏️ Quick check · 2 of 7
Area between curves
The area between two curves is found by integrating:
The difference must be positive.
✍️ Type your answer · 3 of 7
Evaluate ∫ from 1 to 3 of 2x dx.
What is the value?
Enter an exact answer, without rounding.
The antiderivative is x².
✍️ Type your answer · 4 of 7
Evaluate ∫ from 0 to 3 of x² dx.
What is the value?
Enter an exact answer, without rounding.
The antiderivative is x³/3.
✍️ Type your answer · 5 of 7
Evaluate ∫ from 0 to π of sin x dx.
What is the value?
Enter an exact answer, without rounding.
cos π = −1 and cos 0 = 1.
✍️ Type your answer · 6 of 7
Find the average value of f(x) = x² on [0, 3].
What is the average value?
Enter an exact answer, without rounding.
Divide the integral by the interval length.
✍️ Type your answer · 7 of 7
The curve y = x rotates around the x-axis from x = 0 to x = 3. V = π ∫ y² dx.
What is V/π?
Enter an exact answer, without rounding.
Integrate y² = x².
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
The definite integral of a positive function between two limits gives:
Question 2 of 5
∫ from a to a of f(x) dx equals:
Question 3 of 5
Reversing the limits of integration:
Question 4 of 5
If v(t) is velocity, the integral of v from t₁ to t₂ gives:
Question 5 of 5
The trapezium rule is used to:
Lesson 6 · Functions
🌱 Exponentials & Logarithms
Exponential functions describe growth and decay that are proportional to the current amount: money, populations, radioactivity. Logarithms undo them.
🪜 Logarithm laws
logₐ x is the exponent you put on a to get x. The laws turn multiplication into addition.
log₂ 32 = 5 (2⁵ = 32) log₁₀ 1000 = 3
log(ab) = log a + log b
log(a/b) = log a − log b
log(aⁿ) = n log a
ln eᵏ = k
log 2 + log 50 = log 100 = 2
Solve 3ˣ = 81 → x = log₃ 81 = 4
🖼️ See it
The two curves are mirror images in the dashed line y = x, because ln is the inverse of eˣ.
📈 Growth and decay
An exponential model is N = N₀ · bᵗ (or N₀ eᵏᵗ). Doubling time and half-life are the time taken for the amount to double or halve.
P = 500 · 2^(t/3): t = 9 → 500 · 2³ = 4000
Compound interest: 1000 · 1.1² = 1210
Half-life 5 days, 80 g at the start
15 days = 3 half-lives → 80 / 2³ = 10 g
d/dx eˣ = eˣ, and eˣ = e³ → x = 3
Solving for an exponent: take a logarithm of both sides, then use log(aⁿ) = n log a.
✏️ Quick check · 1 of 7
Product law
log(ab) equals:
Think of adding exponents.
✏️ Quick check · 2 of 7
Inverse of exp
The inverse function of eˣ is:
Natural logarithm.
✍️ Type your answer · 3 of 7
Evaluate log₂ 32.
What is the value?
Enter an exact answer, without rounding.
Ask: 2 to what power gives 32?
✍️ Type your answer · 4 of 7
Solve 3ˣ = 81.
What is x?
Enter an exact answer, without rounding.
Write 81 as a power of 3.
✍️ Type your answer · 5 of 7
Simplify log 2 + log 50 (base 10).
What is the value?
Enter an exact answer, without rounding.
Combine into one logarithm.
✍️ Type your answer · 6 of 7
A population is P = 500 · 2^(t/3), with t in hours.
What is P at t = 9?
Enter an exact answer, without rounding.
Compute the exponent first.
✍️ Type your answer · 7 of 7
A sample of 80 g has a half-life of 5 days.
How many grams remain after 15 days?
Enter an exact answer, without rounding.
g
Halve it three times.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
ln(e⁴) equals:
Question 2 of 5
log(aⁿ) equals:
Question 3 of 5
If 2ˣ = 16, then x is:
Question 4 of 5
An investment grows by 10% each year. After 2 years 1000 becomes:
Question 5 of 5
The graph of y = eˣ:
Lesson 7 · Algebra
🌀 Complex Numbers
Complex numbers extend the real numbers with i, where i² = −1. They give every quadratic a solution and describe rotation neatly.
🧩 Arithmetic with i
A complex number is z = a + bi, with real part a and imaginary part b. Add and subtract parts separately, and multiply like brackets, replacing i² by −1.
(3 + 2i) + (1 − 4i) = 4 − 2i
(2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i
Conjugate of a + bi is a − bi
(1 + 2i)(1 − 2i) = 1 + 4 = 5
Division: multiply top and bottom by the conjugate
(4 + 2i)/(1 + i) = (4 + 2i)(1 − i)/2 = (6 − 2i)/2 = 3 − i
Powers of i: i, −1, −i, 1, then repeat; i⁸ = 1
🖼️ See it
A complex number is a point (or arrow) in the plane; the conjugate is its reflection in the real axis. Modulus |z| = √13.
🧭 Modulus, argument and quadratics
On the Argand diagram z is the point (a, b). Its modulus is |z| = √(a² + b²) and its argument is the angle from the positive real axis. Polar form: z = r(cos θ + i sin θ).
Complex roots of real quadratics always come in conjugate pairs.
✏️ Quick check · 1 of 7
i squared
What is i²?
Definition.
✏️ Quick check · 2 of 7
Conjugate
The conjugate of 3 − 4i is:
Only the sign of b changes.
✍️ Type your answer · 3 of 7
(2 + 3i)(1 − i).
What is the real part of the product?
Enter an exact answer, without rounding.
Remember that i² = −1.
✍️ Type your answer · 4 of 7
Find |3 + 4i|.
What is the modulus?
Enter an exact answer, without rounding.
Pythagoras on the Argand diagram.
✍️ Type your answer · 5 of 7
Find |5 + 12i|.
What is the modulus?
Enter an exact answer, without rounding.
Use a² + b².
✍️ Type your answer · 6 of 7
Compute (1 + 2i)(1 − 2i).
What is the result?
Enter an exact answer, without rounding.
Difference of squares.
✍️ Type your answer · 7 of 7
Solve x² + 4x + 13 = 0.
What is the positive imaginary part of a root?
Enter an exact answer, without rounding.
Use the quadratic formula with Δ = −36.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
i⁴ equals:
Question 2 of 5
The Argand diagram plots a complex number a + bi at the point:
Question 3 of 5
The argument of 1 + i is:
Question 4 of 5
For a quadratic with negative discriminant the roots are:
Question 5 of 5
(1 + i)² equals:
Lesson 8 · Data
📊 Statistics & Inference
Statistics lets us draw careful conclusions about a whole population from a sample, and say how confident we are in them.
🔔 The normal distribution
For a normal distribution with mean μ and standard deviation σ, the z-score z = (x − μ)/σ says how many standard deviations x is from the mean. About 68% of values lie within 1σ, 95% within 2σ and 99.7% within 3σ.
x = 70, μ = 60, σ = 5 → z = (70 − 60)/5 = 2
Binomial, n = 100, p = 0.5
mean np = 50, sd √(npq) = √25 = 5
Expected value E(X) = Σ x·p
values 1, 2, 3 with p 0.2, 0.5, 0.3:
0.2 + 1.0 + 0.9 = 2.1
🖼️ See it
A 95% confidence interval spans about ±1.96 standard errors around the sample mean.
🔍 Estimating and testing
The standard error of a sample mean is σ/√n. A 95% confidence interval is x̄ ± 1.96·SE. In a hypothesis test compare z = (x̄ − μ₀)/SE to the critical value 1.96 (5% level, two-tailed).
σ = 20, n = 100 → SE = 20/10 = 2
x̄ = 50 → margin = 1.96·2 = 3.92
95% CI = 50 ± 3.92 → upper bound 53.92
H₀: μ = 50, sample mean 55
z = (55 − 50)/2 = 2.5 > 1.96 → reject H₀
Regression line y = 2x + 1 at x = 4 → 9
The p-value is the probability of a result at least this extreme if H₀ is true. It is not the probability that H₀ is true.
✏️ Quick check · 1 of 7
Empirical rule
In a normal distribution, about what percentage of values lie within 2 standard deviations of the mean?
Remember the three numbers.
✏️ Quick check · 2 of 7
p-value
A small p-value (below 0.05) means:
It measures how surprising the data is.
✍️ Type your answer · 3 of 7
A test score is 70. The mean is 60 and the standard deviation is 5.
What is the z-score?
Enter an exact answer, without rounding.
z = (x − μ) / σ.
✍️ Type your answer · 4 of 7
A population has σ = 20 and a sample has n = 100.
What is the standard error of the mean?
Enter an exact answer, without rounding.
SE = σ / √n.
✍️ Type your answer · 5 of 7
The sample mean is 50 with standard error 2.
What is the upper end of the 95% confidence interval?
Enter an exact answer, without rounding.
Add the margin of error 1.96·SE.
✍️ Type your answer · 6 of 7
H₀: μ = 50. A sample gives x̄ = 55 with standard error 2.
What is the test statistic z?
Enter an exact answer, without rounding.
z = (x̄ − μ₀) / SE.
✍️ Type your answer · 7 of 7
X takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3.
What is E(X)?
Enter an exact answer, without rounding.
Sum value times probability.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The standard error decreases when the sample size:
Question 2 of 5
A binomial distribution has n = 100 and p = 0.5. The mean is:
Question 3 of 5
A correlation coefficient r always lies between:
Question 4 of 5
A 95% confidence interval means that:
Question 5 of 5
Correlation does not imply:
Final Challenge
🏆 16-Question Challenge Quiz
Questions from every topic in this course. Pick an answer for each one, then hit Check My Answers.
0 / 16
💪 Keep practicing — go back through the lessons and retake the quiz!
Q1 · Limits & Continuity
For lim (x² − 9)/(x − 3) as x→3, the first step is to:
Q2 · Limits & Continuity
The slope of a secant line becomes the slope of the tangent when: