Grade 12 · MCV4U · Calculus & Vectors

MCV4U Formula Sheet

Q: Do you get a formula sheet on the MCV4U exam?

That is up to your teacher. Some hand one out, some let you bring your own page, some give none. Ask early, and learn the formulas either way: knowing which one to use is half the question.

Q: Which MCV4U formulas should I memorize?

The derivative rules, the derivatives of e^x, ln x, sin x and cos x, the dot and cross products, and the scalar equation of a plane. They show up in almost every question in their units.

Every formula from Calculus and Vectors, grouped by unit, in the order you learn them. Print the PDF for your binder or keep this page open while you practise. No signup needed.

Download the PDF MCV4U practice exam

MCV4U Formula Sheet · Grade 12 Calculus & VectorsOntario curriculum

Limits

Average rate of change: $\frac{f(b) - f(a)}{b - a}$
Slope of the tangent: $m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$
Limit laws: $\lim_{x \to a}\big[f(x) \pm g(x)\big] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$
Products, quotients (nonzero denominator) and constant multiples work the same way.
Limit exists: $\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$
Continuity at a point: $\lim_{x \to a} f(x) = f(a)$
Limits at infinity: $\lim_{x \to \pm\infty} \frac{1}{x^n} = 0$
For $n > 0$ . Divide by the highest power of $x$ in the denominator.

Derivatives

First principles: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
Constant and power rules: $\frac{d}{dx}(c) = 0 \qquad \frac{d}{dx}\left(x^n\right) = nx^{n-1}$
Constant multiple and sum: $\big(cf\big)' = cf' \qquad \big(f \pm g\big)' = f' \pm g'$
Product rule: $\big(fg\big)' = f'g + fg'$
Quotient rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$
Chain rule: $\frac{d}{dx}f\big(g(x)\big) = f'\big(g(x)\big)\,g'(x) \qquad \frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$
Tangent line: $y - f(a) = f'(a)(x - a)$
The normal line has slope $-\frac{1}{f'(a)}$ .
Motion: $v(t) = s'(t) \qquad a(t) = v'(t) = s''(t)$
Business: $R(x) = x\,p(x) \qquad P(x) = R(x) - C(x)$
Marginal cost, revenue and profit are $C'(x)$ , $R'(x)$ and $P'(x)$ .
Implicit differentiation: $\frac{d}{dx}\left(y^n\right) = ny^{n-1}\frac{dy}{dx}$

Curve Sketching

Increasing and decreasing: $f'(x) > 0 \Rightarrow f \text{ increasing} \qquad f'(x) < 0 \Rightarrow f \text{ decreasing}$
Critical number: $f'(c) = 0 \ \text{ or } \ f'(c) \text{ undefined}$
Concavity: $f''(x) > 0 \Rightarrow \text{up} \qquad f''(x) < 0 \Rightarrow \text{down}$
A point of inflection is where $f''$ changes sign.
Second derivative test: $f''(c) > 0 \Rightarrow \text{local min} \qquad f''(c) < 0 \Rightarrow \text{local max}$
At a critical number $c$ with $f'(c) = 0$ .
Absolute extrema on [a, b]: $\text{compare } f(a),\ f(b),\ f(c)$
$c$ is each critical number in $[a, b]$ ; the largest value is the max, the smallest the min.

Optimization

Optimize: $f'(x) = 0$
Then check critical numbers and endpoints of the domain.
Related rates: $\frac{dV}{dt} = \frac{dV}{dr}\cdot\frac{dr}{dt}$
Differentiate the equation linking the quantities with respect to $t$ .
Cylinder: $V = \pi r^2 h \qquad SA = 2\pi r^2 + 2\pi rh$
Cone: $V = \frac{1}{3}\pi r^2 h$
Sphere: $V = \frac{4}{3}\pi r^3 \qquad SA = 4\pi r^2$

Derivatives of Trig, Exp and Log

Exponential: $\frac{d}{dx}e^x = e^x \qquad \frac{d}{dx}b^x = b^x\ln b$
Logarithmic: $\frac{d}{dx}\ln x = \frac{1}{x} \qquad \frac{d}{dx}\log_b x = \frac{1}{x\ln b}$
Chain rule with e: $\frac{d}{dx}e^{g(x)} = e^{g(x)}\,g'(x)$
Sine and cosine: $\frac{d}{dx}\sin x = \cos x \qquad \frac{d}{dx}\cos x = -\sin x$
Tangent: $\frac{d}{dx}\tan x = \sec^2 x$
Key limit: $\lim_{x \to 0}\frac{\sin x}{x} = 1$
$x$ in radians.

Geometric Vectors

Resultant: $\vec{R} = \vec{u} + \vec{v}$
Tip to tail, or the diagonal of the parallelogram.
Magnitude of a sum: $|\vec{u} + \vec{v}|^2 = |\vec{u}|^2 + |\vec{v}|^2 + 2|\vec{u}||\vec{v}|\cos\theta$
$\theta$ is the angle between $\vec{u}$ and $\vec{v}$ placed tail to tail.
Equilibrant: $\vec{E} = -\vec{R}$
Unit vector: $\hat{u} = \frac{\vec{u}}{|\vec{u}|}$
Ground velocity: $\vec{v}_{\text{ground}} = \vec{v}_{\text{air}} + \vec{v}_{\text{wind}}$

Cartesian Vectors

Magnitude: $|\vec{u}| = \sqrt{u_1^2 + u_2^2 + u_3^2}$
Vector between points: $\overrightarrow{AB} = \left(x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1\right)$
Dot product: $\vec{u}\cdot\vec{v} = |\vec{u}||\vec{v}|\cos\theta = u_1v_1 + u_2v_2 + u_3v_3$
$\vec{u}\cdot\vec{v} = 0$ when the vectors are perpendicular.
Scalar projection of u on v: $|\vec{u}|\cos\theta = \frac{\vec{u}\cdot\vec{v}}{|\vec{v}|}$
Vector projection of u on v: $\text{proj}_{\vec{v}}\vec{u} = \left(\frac{\vec{u}\cdot\vec{v}}{\vec{v}\cdot\vec{v}}\right)\vec{v}$
Cross product: $\vec{u}\times\vec{v} = \left(u_2v_3 - u_3v_2,\ u_3v_1 - u_1v_3,\ u_1v_2 - u_2v_1\right)$
Magnitude of the cross product: $|\vec{u}\times\vec{v}| = |\vec{u}||\vec{v}|\sin\theta$
The area of the parallelogram the vectors span; half of it for the triangle.
Work and torque: $W = \vec{F}\cdot\vec{d} \qquad |\vec{\tau}| = |\vec{r}\times\vec{F}| = |\vec{r}||\vec{F}|\sin\theta$
Coplanar vectors: $\vec{u}\cdot\left(\vec{v}\times\vec{w}\right) = 0$

Lines and Planes

Line, vector form: $\vec{r} = \vec{r}_0 + t\vec{m},\ t \in \mathbb{R}$
Line, parametric form: $x = x_0 + tm_1 \qquad y = y_0 + tm_2 \qquad z = z_0 + tm_3$
Line, symmetric form: $\frac{x - x_0}{m_1} = \frac{y - y_0}{m_2} = \frac{z - z_0}{m_3}$
For $m_1, m_2, m_3 \neq 0$ .
Line in 2D, scalar form: $Ax + By + C = 0$
Normal $\vec{n} = (A, B)$ .
Plane, vector form: $\vec{r} = \vec{r}_0 + s\vec{a} + t\vec{b},\ s, t \in \mathbb{R}$
Plane, scalar form: $Ax + By + Cz + D = 0$
Normal $\vec{n} = (A, B, C) = \vec{a}\times\vec{b}$ .
Distance from a point to a plane: $d = \frac{\left|Ax_0 + By_0 + Cz_0 + D\right|}{\sqrt{A^2 + B^2 + C^2}}$
Distance from a point to a 2D line: $d = \frac{\left|Ax_0 + By_0 + C\right|}{\sqrt{A^2 + B^2}}$
Distance between skew lines: $d = \frac{\left|\overrightarrow{P_1P_2}\cdot\left(\vec{m}_1\times\vec{m}_2\right)\right|}{\left|\vec{m}_1\times\vec{m}_2\right|}$

MCV4U Formula Sheet FAQ

Do you get a formula sheet on the MCV4U exam?+

That is up to your teacher. Some hand one out, some let you bring your own page, some give none. Ask early, and learn the formulas either way: knowing which one to use is half the question.

Which MCV4U formulas should I memorize?+

The derivative rules, the derivatives of e^x, ln x, sin x and cos x, the dot and cross products, and the scalar equation of a plane. They show up in almost every question in their units.

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