Limits, derivatives, integrals and differential equations

Grade 12 Calculus

Built to BC Calculus 12. Also covers Alberta Mathematics 31 and the calculus in Ontario's MCV4U.

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Limits & Continuity

BC Calculus 12Alberta Mathematics 31Ontario MCV4U

L1
Lesson 1
Pieces of a function
Evaluate by piece · |x| is piecewise
A piecewise function is one function written with more than one rule, each rule in charge of part of the number line.
Start
L2
Lesson 2
Undoing sine
arcsin, arccos, arctan · restricted domains and ranges
Saying arcsine out loud is the cheapest insurance in this chapter.
Start
L3
Lesson 3
Closing in
A limit from a table and from a graph
Here f is the function, a is the input being approached, and L is the number the outputs close in on as the inputs get close to a.
Start
L4
Lesson 4
Factor, cancel, substitute
0/0 is a question, not an answer
It is not zero, not one, not undefined, not a verdict of any kind: it is the form telling you that N and D share a factor (x - a), which is the only way both can vanish at the same place.
Start
L5
Lesson 5
Left, right, both
The two-sided limit exists only when both sides agree
If they disagree, the two-sided limit does not exist; there is no averaging, no tie-break, no side that wins. A jump is precisely this: two one-sided limits at different heights.
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L6
Lesson 6
Where it ends up
End behaviour and horizontal asymptotes
For a quotient of polynomials, only the highest power on each side survives far out, so compare the degrees (the highest powers).
Start
L7
Lesson 7
No lifting the pencil
Three conditions · removable, jump and infinite discontinuities
A function is continuous at a when you can draw its graph through x = a without lifting the pencil.
Start
L8
Lesson 8
A root must be there
The IVT on a sign change
If f is continuous on the closed interval [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.
Start
★
Boss lesson
The Gap Inspection
the boss · no calculator · one piecewise function, every question
No calculator: the function is three lines, a square plus a constant on the left, a lone value at the boundary, a line on the right, and every number is small enough to hold in your head.
Start

The Derivative

BC Calculus 12Alberta Mathematics 31Ontario MCV4U

L10
Lesson 10
From secant to tangent
Average rate over a shrinking interval
A secant line cuts a curve at two points.
The formulas in this lesson
$m = \dfrac{f(a + h) - f(a)}{h}$ , $m = \dfrac{f(b) - f(a)}{b - a}$ , $v_{\text{avg}} = \dfrac{s(t_2) - s(t_1)}{t_2 - t_1}$
Start
L11
Lesson 11
The limit definition
f′(x) = lim (f(x + h) − f(x))/h · polynomials from scratch
The slope of the tangent at every x at once is a new function, the derivative of f, written f'(x) and read aloud f prime of x.
Start
L12
Lesson 12
Four ways to write it
f′(x), dy/dx, d/dx, and what each says
The derivative has four common notations and an exam uses all of them. f'(x), f prime of x, names the derivative as a function; f'(3) is that function at 3, a plain number, the slope at x = 3.
Start
L13
Lesson 13
The power rule
d/dx xⁿ = n xⁿ⁻¹, with constants and sums
Here n is the exponent, any fixed real number, and the rule does two things in one move: the old exponent comes down in front as a factor, and the exponent itself drops by one.
The formulas in this lesson
$\dfrac{d}{dx}\,x^{n} = n\,x^{n-1}$ , $\sqrt{x} = x^{1/2}$ , $\dfrac{d}{dx}\,c = 0$ , $\dfrac{1}{x} = x^{-1}$
Start
L14
Lesson 14
The tangent line
Linear approximation
The tangent line to f at x = a passes through the point (a, f(a)) with slope f'(a). Its equation is y = f(a) + f'(a)(x - a), read aloud: y equals f of a, plus f prime of a times x minus a.
The formulas in this lesson
$y = f(a) + f'(a)(x - a)$ , $f(a + \Delta x) \approx f(a) + f'(a)\,\Delta x$
Start
L15
Lesson 15
Rates of rates
f″ and f‴ · position, velocity, acceleration
f' is a function, so it has a derivative of its own: the second derivative f''(x), read aloud f double prime of x, is the derivative of f'(x).
Start
L16
Lesson 16
Speed at an instant
v = s′(t) from a position polynomial
A position function s(t) gives where an object is, in metres, at each time t in seconds.
The formula in this lesson
$v(t) = s'(t)$
Start
★
Boss lesson
The Speed Trap
the boss · no calculator · one position function, every question
Write each line down as you go; a slip in line two walks all the way to line seven. 5★ under par is the chapter crown.
The formulas in this lesson
$v(t) = s'(t)$ , $a(t) = v'(t)$ , $s = s(t_0) + v(t_0)(t - t_0)$
Start

Differentiation Rules

BC Calculus 12Alberta Mathematics 31Ontario MCV4U

L18
Lesson 18
The product rule
(fg)′ = f′g + fg′
Module 2 differentiated sums term by term and pulled constants out in front. A product of two functions of x is the first thing that refuses to be done piece by piece.
The formula in this lesson
$(fg)' = f'g + fg'$
Start
L19
Lesson 19
The quotient rule
(f/g)′ = (f′g − fg′)/g²
The two terms on top are the product rule's two terms with the sign between them changed, and the bottom is squared.
The formula in this lesson
$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$
Start
L20
Lesson 20
The chain rule
Outside, then inside · dy/dx = dy/du · du/dx
A change in x is scaled by the inside, then scaled again by the outside, and scalings multiply; on paper the du's cancel as if they were numbers, which is how to remember the form.
The formulas in this lesson
$\dfrac{d}{dx}\,u^{n} = n\,u^{n-1} \cdot \dfrac{du}{dx}$ , $\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}$ , $\dfrac{d}{dx}\,\sqrt{u} = \dfrac{du/dx}{2\sqrt{u}}$
Start
L21
Lesson 21
Exponentials and logs
d/dx eˣ = eˣ, aˣ ln a, 1/x
Anything to the power 0 is 1, not 0 and not e.
The formulas in this lesson
$\dfrac{d}{dx}\,e^{x} = e^{x}$ , $\dfrac{d}{dx}\,e^{bx} = b\,e^{bx}$ , $\dfrac{d}{dx}\,\ln x = \dfrac{1}{x}$ , $\dfrac{d}{dx}\,a^{x} = a^{x}\ln a$
Start
L22
Lesson 22
Trig derivatives
sin, cos, tan and their chain forms, in radians
The second trap is degrees.
The formulas in this lesson
$\dfrac{d}{dx}\,\sin x = \cos x$ , $\dfrac{d}{dx}\,\sin(bx) = b\cos(bx)$ , $\dfrac{d}{dx}\,\cos x = -\sin x$ , $\dfrac{d}{dx}\,\cos(bx) = -b\sin(bx)$
Start
L23
Lesson 23
Implicit differentiation
dy/dx without solving for y
Implicit differentiation skips the solving: treat y as a function of x that happens not to be written out, and differentiate both sides with respect to x.
Start
L24
Lesson 24
Which rule first
Reading a function's shape to choose the rules
Five rules now, and the exam's real question is which one opens a given function. The answer is the outermost operation: the LAST thing done in building f.
The formulas in this lesson
$2\sec^2(2x)$ , $\dfrac{2x}{x^2 + 1}$ , $2\cos(2x)$ , $2\sin x\cos x$
Start
★
Boss lesson
The Rule Book
the boss · no calculator · one composite function
Write each line down as you go; a slip in line four walks all the way to line seven. 5★ under par is the chapter crown.
The formulas in this lesson
$(fg)' = f'g + fg'$ , $\dfrac{d}{dx}\,u^{n} = n\,u^{n-1} \cdot \dfrac{du}{dx}$ , $y = f(a) + f'(a)(x - a)$
Start

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