# MCV4U – Grade 12 Calculus & Vectors – Exam Notes

Grade 12 – Calculus and Vectors

Geometric Vectors Test

Vectors

• Vector: is a quantity that has direction and magnitude
• Scalar: is a quantity that only has magnitude
• True Bearing: is directed compass measurement, beginning at North and rotating clockwise.
• Quadrant Bearing: is a compass measurement east or west of the North-South line
• Equivalent Vectors: are equal in magnitude and direction.
• Vectors can be translated anywhere on the same plane and still be equivalent
• Oppose Vectors: are equal in magnitude, but opposite in direction
• Vectors can be

Adding and Subtracting Vectors

• Since vectors remain equivalent no matter where they are translated, they can be moved around to construct diagrams more convenient for solving.
• Adding Vectors using Tail to head method using a triangle or tail to tail method using a parallelogram.
• The resultant is a vector joining the head of the first vector to the tail of the last vector
• If 2 vectors, a and b, are parallel in the same direction, |a + b| = |a| + |b| and in the same direction
• If a and b have opposite directions and | a | > | b |, then | a + b | = | a | – | b | and a + b is in the same direction as a.
• Subtract vectors by adding the opposite vector
• Zero Vector: means there is no magnitude or direction. Addition of 2 opposite vectors
• Vectors follow commutative, associative, and identity properties
• They can be added in any order
• Simplifying vector expressions is similar to simplifying integer expressions

Scalar Vector Multiplication

• When a vector is multiplied by a scalar, the magnitude is multiplied by the scalar and the vectors are parallel. The directions remains unchanged if the scalar is positive, and becomes opposite if scalar is negative.
• Multiplying vectors follow the distributive and associative rule
• They can be expanded using FOIL
• They can be multiplied in any order
• Linear Combination of vectors can be formed by adding scalar multiples of 2 or more vectors.

Applications of Geometric Vectors

• 2 Vectors that are perpendicular to each other and add together to give a vector v are called the rectangular vector components of v.
• When solving resultants, you can use Vector operations, pythagorean theorem, or trigonometry.
• Equilibriant Vector: is the opposite of the resultant

Grade 12 – Calculus and Vectors

Cartesian Vectors Test

Vectors

• Unit Vectors: are i = [1, 0], j = [0,1] have magnitude 1 and tails at origin.
• Cartesian Vector is a representation of a vector on the Cartesian plane where the endpoints are the points on the Cartesian plane.
• If a vector, u, is translated so that the tail is at 0,0, then that vector is called a position vector. Position vectors are represented in square brackets where the points are [u1 , u2].
• Magnitude of u = [u1, u2] is |u| = √(u12 + u22)
• Horizontal and vertical components of vector u can be said as [u1, 0] and [0, u1] respectively
• For vectors u = [u1, u2] and v = [v1, v2] and scalar k,
• u + v = [u1 + v1, u2 + v2]
• u – v = [u1 – v1, u2 – v2]
• kv = [kv1, kv2]
• Cartesian vector between 2 points P1(x1, y1) and P2 (x2, y2) is P1P2 = [x2 – x1, y2 – y1]
• Geometric vectors can also be written as:
• v = [ |v| cosX , |v| sinX ] where X is the angle v makes with the positive x-axis

Dot Product

• The dot product is defined as a∙b = |a| |b| cos X, where X is the angle between a and b.
• Dot product produces a scalar.
• For any vectors u, v, and w and scalar k,
• u and v cannot be zero are perpendicular if and only if u∙v = 0
• Commutative property
• Associative property
• Distributive property
• u u = |u|2
• u ∙ 0 = 0
• If u = [u1, u2] and v = [v1, v2] then u∙v = u1*v1 + u2*v2

Applications of Dot Product

• For any 2 vectors u and v with angle of X between them, the projection of v on u is the vector component of v in the direction of u.
• proj u v = |v| cos X (1/|u| *|u|) or proj u v = (v∙u / u∙u) u
• |proj u v |= |v| cos X  if 0 < X < 90
• |proj u v |= – |v| cos X  if 90 < X < 180
• |proj u v |= | u∙v / |u| |
• Based on the angle between u and v, you can determine what direction the projection will be.

Vectors in 3 space

• Same properties and formulas as Vectors in 2 space
• Orthogonal: If 2 vectors are orthogonal, the angle between them is 90 degrees.

Cross Product

• The cross product between 2 vectors will find a vector that is perpendicular between both of them
• Used to find torque of it
• Direction is found by using right hand rule. Fingers point towards vector r, then bend towards vector f, thumb points in direction of the cross product
• a X b = ( |a| |b| sin X)n where n is the unit vector orthogonal to both a and b following the right hand rule for direction and X is the angle between the vectors.
• For cartesian vectors:
• If v = [v1, v2, v3] and u = [u1, u2, u3]
• u X v = [u2v3 – u3v1 , u3v1 – u1v3 , u1v2 – u2v1]
• |a X b| = |a| |b| sin X
• This also calculates the area of a parallelogram
• Properties
• u X v = -(v X u)
• distributive property
• associative property
• commutative property
• If u != 0 and v != 0, u X v = 0 if and only if u = mv, in other words, they must be collinear for that to happen.

Applications of Cross Product

• Torque is the cross product between the length of the wrench and the force applied
• Formulas for projection work in 3D and 2D
• Triple Scalar Product: a * b X c
• The cross product must be done before the dot product is done
• The volume of a parallelogram is: V = |w * u X v |
• Work against gravity is the dot product only with the z axis for gravity.

Grade 12 – Calculus and Vectors

Lines and Planes Test

Equation of 2-Space and 3-Space lines

• In 2-Space, a line can be defined in 4 ways: Slope y-intercept form, vector equation, parametric form, or scalar form
• Slope Y-Intercept form: y = mx + b
• m is the slope
• b is the y intercept
• Vector Form: r = r0+ tm
• or: [x , y] = [x0 , y0] + t[m1 , m2]
• [x,y] is a position vector to any point on the line
• [x0 , y0] is a position vector to any point on the line
• [m1 , m 2] is the direction vector for the line
• Parametric Form:
• x = xo + tm1 or x = x0 + t[m1]
• y = y0 + tm2 or y = y0 + t[m2]
• Same definitions as Vector form, derived from vector form
• Scalar Form: ax + by + c = 0
• n = [a, b] is a normal vector to the line
• If direction vector d = [a, b], then normal vector n = [-b, a]

• In 3-Space, a line can be defined by a vector equation or parametric equation
• Vector Form: r = r0+ tm
• or: [x , y , z] = [x0 , y0 , z0] + t[m1 , m2 , m3]
• [x,y,z] is a position vector to any point on the line
• [x0 , y0 , z0] is a position vector to any point on the line
• [m1 , m2 , m3] is the direction vector for the line
• Parametric Form:
• x = xo + tm1 or x = x0 + t[m1]
• y = y0 + tm2 or y = y0 + t[m2]
• z = z0 + tm3 or y = z0 + t[m3]
• Same definitions as Vector form, derived from vector form
• A normal vector is a line perpendicular to that line
• To determine a line, 2 points, or a point and a direction vector are needed
• For lines in 2-Space, a point and the normal vector can also determine the line

Equations of Planes

• In 2-Space, a scalar equation defines a line. In 3-space, a scalar equation defines a plane
• In 3-Space, a plane can be defined by a vector equation, parametric equation, or a scalar equation
• Vector Form: r = r0+ tm
• or: [x , y , z] = [x0 , y0 , z0] + t[a1 , a2 , a3] + s[b1 , b2 , b3]
• [x,y,z] is a position vector to any point on the plane
• [x0 , y0 , z0] is a position vector to any point on the plane
• [a1 , a2 , a3] and [b1 , b2 , b3] have to be non-parallel vectors on the plane
• Parametric Form:
• x = xo + ta1 + tb1    or x = x0 + t[a1] + s[b1]
• y = y0 + ta2 + tb2   or y = y0 + t[a2] + s[b2]
• z = z0 + ta3 + tb3   or y = z0 + t[a3] + s[b3]
• Same definitions as Vector form, derived from vector form
• Scalar Form: ax + by + cz + d = 0
• Where [a,b,c] is a vector normal to the plane
• A plane can be defined by 3 non-collinear points or a point and 2 non-parallel direction vectors
• Intercepts of a plane
• The X-Intercept of a plane is found by setting y = z = 0 and solving for x
• The Y and Z intercepts are the same by setting the other 2 variables to zero

Intersection of Lines in 2-Space and 3-Space

• In 2-Space, the intersection of 2 lines have 3 possibilities.
• (1) Lines intersect at exactly one point
• (2) Lines are coincident, so there are infinitely many solutions
• (3) Lines are parallel, no solutions
• In 3-Space, the intersection of 2 lines can have 4 possibilities
• (1) They intersect at one point
• (2) They are coincident
• (3) They are parallel
• (4) They are skewed lines, where they are not parallel and have no solution
• To find the distance between 2 skew lines, the formula:
• d = | (P1P2 dot n / n) |
• where P1 and P2 are any points on each line and n = m1 x m2.
• To find the intersection, simply equate each pair of lines in parametric form and solve for X, Y, and Z in each set.

Intersection of Planes and Lines

• In 3-Space, the intersection of planes and lines have 3 possibilities
• (1) The line intersects right through the plane at one point
• (2) The line is coincident with the plane at infinitely many points
• (3) The line is parallel and not touching the plane with no solutions
• The distance between between point P and a plane is:
• d = |n dot PQ| / |n|
• where n is a normal vector to the plane, Q is any point on the plane
• To find the intersection, substitute each X, Y and Z of the parametric equation into the equation of the plane. Then after t is found, substitute that back into the line to find what each value of X Y and Z are.

Intersection of Planes

• There are 3 possibilities for the intersection of 2 planes
• (1) The planes intersect at a line
• (2) The planes are coincident, with infinitely many solutions
• (3) The planes are parallel with no solutions
• With 2 planes:
• 1) Check if the normals are parallel
• 2) Try solving based on whether or not the normals are parallel
• If they are parallel, you’ll get no or infinite solutions
• If they aren’t parallel, you’ll get a line as the solution

• There are 7 possibilities for intersection of 3 planes:
• If their normals are parallel
• (1) All 3 planes are coincident if normals are all the same
• (2) All 3 planes are parallel if normals are all distinct
• (3) 2 Planes are coincident and 1 is parallel if 2 normals are distinct
• (4) 1 Plane intersects 2 planes that are coincident or parallel if 2 normals are the same
• If their normals are not parallel
• If triple scalar product (n1 x n2)*n3 = 0
• (5) A Triangular prism is formed and no intersection (0x + 0y = c)
• (6) A “revolving door” is formed with a line intersection (0x + 0y = 0)
• If triple scalar product is not = 0
• (7) All planes intersect at a point
• For revolving door intersection and intersection of 2 planes at a line, a parameter z = t should be used to find the line of intersection.

Grade 12 – Calculus and Vectors

Rates of Change Test

Rates of Change and the slope of a curve

• Average rate of change: refers to the rate of change of a function over an interval. It corresponds to the slope of the secant connecting the two endpoints of the interval
• Instantaneous rate of change: refers to the rate of change at a specific point. It corresponds to the slope of a tangent passing through a single point, or tangent point, on the graph of a function.
• An estimate of the instantaneous rate of change can be obtained by calculating the average rate of change over the smallest interval for which data are available.
• It can also be determined by sketching a tangent on a graph

Rates of Change using Equations

• For a given function y = f(x), the instantaneous rate of change at x = a is estimated by calculating the slope of a secant over a very small interval, a <= x <= a + h such that h is a very small number
• The expression f(a+h) – f(a) / h, where h not = 0, is called the difference quotient
• It is used to calculate the slope of the secant between (a, f(a)) and (a + h, f(a + h)). It will let you have an estimate for the slope of the tangent as h approaches 0.

Limits

• A sequence is a function, f(n) = tn , whose domain is the set of natural numbers N
• limit as x approaches a of f(x) exists if the following are true:
• limit as x approaches a- of f(x) exists (left hand limit exists)
• limit as x approaches a+ of f(x) exists (right hand limit exists)
• right side limit and left side limit are equal
• If they aren’t equal and they criteria aren’t met, then the limit does not exists
• Limits can be a real number of infinity
• A function is continues at a value if these are true:
• f(a) exists
• limit as x approaches a exists
• The limit as x approaches a and f(a) are the same

Limits and Continuity

• The limit of a function at x = a may exist even though the function is discontinuous at x = a.
• In short, the graph of a discontinuous function cannot be drawn without lifting your pencil
• There are 3 kinds if discontinuity: Jump, Removable, and Infinite
• When direct substitution of x = a results in a limit of an indeterminate , 0/0, determine an equivalent function that represents f(x).
• Discontinuity may be removed by factoring, rationalizing the numerator or denominator, expanding, and simplifying.

Introduction to derivatives

• The derivatives of y = f(x) is a new function y = f’(x), which represents the slope of the tangent, or instantaneous rate of change, at any point on the curve of y = f(x).
• The derivative function is defined by the first principles definition for the derivative, f’(x) = lim h -> 0   f(x+h) – f(x) / h, if the limit exists
• Different notations for the derivative of y = f(x) are y = f’(x), y’, dy/dx, and d/dx f(x).
• If the derivative does not exist on a point on the curve, the function is non-differentiable at that x-value. This can occur at points where the function is discontinuous or in cases where the function has an abrupt change, which is represented by a cusp or corner on the graph.

Grade 12 – Calculus and Vectors

Derivatives Test

Derivatives of a Polynomial Function

• Derivative rules simplify the process of differentiating polynomials with first principles
• When differentiating radicals, we rewrite radicals in fraction exponent form
• ie root x = x^1/3
• To differentiate a power of x that is in the denominator, first express it as a power with a negative exponent
• Derivative rule: xn = nxn-1
• Sum of derivatives: (f(x) + g(x))’= f’(x) + g’(x)
• Difference of derivatives: (f(x) – g(x))’ = f’(x) – g’(x)

Product Rule

• Product Rule: (f(x) * g(x)) = f’(x)g(x) + f(x)g’(x)
• Leibniz Notation: d/dx[f(x)g(x)] = d/dx[f(x)]g(x) + f(x) d/dx[g(x)]

Velocity, Acceleration, and Second Derivatives

• The second derivative of a function is determined by differentiating the first derivative of the function
• For a given position function s(t), its velocity function is v(t), or s’(t), and its acceleration is a(t), v’(t), or s’’(t)
• When v(t) = 0, the object is at rest. There are many instances when an object will be temporarily be at rest when changing directions.
• When v(t) > 0, the object is moving in the positive direction
• When v(t) < 0, the object is moving in the negative direction
• When a(t) > 0, the velocity of the object is increasing
• When a(t) < 0, the velocity of the object is decreasing
• An object is speeding up if a(t) x v(t) > 0 and slowing down if v(t) x a(t) < 0.

Chain Rule

• Used to differentiate composite functions, f = g o h.
• Given a function, the Chain rule is:
• (f(g(x)))’ = f’(g(x) * g’(x)
• In Leibniz notation,
• dy/dx = dy/du * du/dx

Quotient Rule

• To find the derivative of a quotient:
• q(x) = f’(x)g(x) – f(x)g’(x) / g2(x)

Rate of Change problems

• Demand or price function p(x) is the price at which x units of a product or service can be sold
• Revenue function R(x) is the total revenue from the sale of x units of a product or service. R(x) = x * P(x)
• Cost function, C(x), is the total cost of producing x units of a product or service
• Profit function, P(x) is P(x) = R(x) – C(x)
• C’(x) is the marginal cost function
• R’(x) is the marginal revenue function
• P’(x) is the marginal profit function

Grade 12 – Calculus and Vectors

Curve Sketching Test

Interval of Increase and Decrease

• A function is increasing if the slope of the tangent is positive over the entire interval
• A function is decreasing if the slope of the tangent is negative over the entire interval
• Intervals over which a function is increasing or decreasing can be determined by finding the derivative f’(x), and solving inequalities f’(x) <= 0 or f’(x) >= 0.

Maxima and Minima

• If f’(x) changes from positive to zero to a negative, then whenever it’s 0 will be a local maximum point
• If f’(x) changes from negative to zero to a positive, then whenever it’s 0 will be a local minimum point
• The absolute maximum and minimum values are found at local extrema or at the endpoints of the interval
• A critical number is a number in the domain of the function where f’(a) = 0 or f’(a) exists

Concavity and the second derivatives test

• The second derivative is the derivative of the first derivative. It is the rate of change of the slope of the tangent
• A function is concave up on an interval if the second derivative is positive on that interval
• A function is concave down on an interval if the second derivative is negative
• A function has a point of inflection at the point where the second derivative changes sign, where f’’(x) = 0
• Critical points can be classified by using second derivative or by examining the graph of f’’(x)
• If f’(a) = 0 and f’’(a) > 0, there is a local minimum at (a, f(a))
• If f’(a) = 0 and f’’(a) < 0, there is a local maximum at (a, f(a))
• If f’(a) = 0 and f’’(a) = 0 and f’’(x) changes sign at x = a, then there is a POI at (a, f(a))

Simple rational functions

• Vertical Asymptotes are the values of x that make the denominator equal to zero
• Consider VAs when finding intervals of concavity of increase or decrease

Curve Sketching

• When sketching a curve, remember to include domain, asymptotes, intercepts, critical values from the first derivative, second derivative critical values, local and absolute extremas, points of inflection, intervals of increase and decrease, and concavity.

Optimization problems

• Identify what the problem is asking
• Define variables and create a diagram if it helps
• Identify the quantity to be optimized and write an equation
• Define independent variable
• Define a function
• Identify and restrictions on the function
• Differentiate the function and classify critical points
• Check if the solution is whatever the question asked and if its in context with the question

Grade 12 – Calculus and Vectors

Sinusoidal and Logarithmic Derivatives

Derivatives Sine and Cosine functions

• Derivative of y = sin x is y’ = cos x
• Derivative of y = cos x is y’ = -sin x
• Derivative of y = sin (f(x)) is y’ = cos (f(x)) * f’(x)
• Derivative of y = cos (f(x)) is y’ = -sinx (f(x)) * f’(x)
• Derivative of y = sin2 (f(x)) is y’ = 2 cos (f(x)) * f’(x)
• Derivative of y = cos2 (f(x)) is y’ = -2 sin (f(x)) * f’(x)

Simple Harmonic Motion Trigonometric Application Problems

• 1st Derivative used to find velocities
• 2nd Derivatives used to find accelerations as well as max/min velocities
• To find period from its equation, it’s 360/k or 2 Pi / k for radians

The number e

• The symbol e is defined as limit when n -> infinity (1 + 1/n)n . The value is ~ 2.71
• Rate of change of exponential function is also exponential
• Derivative of y = ex is y = ex (the same as original function)

Natural Logarithm

• Lnx = logex
• The functions y = lnx and y = ex are inverses

Derivatives of exponential functions

• The derivative of y = bx is y’ = bx * lnx
• The derivative of y = bf(x) is y’ = bf(x) * ln b * f’(x)
• The derivative of y = ef(x) is y’ = ef(x) * f’(x)
• You solve most logarithms by applying ln or log both sides and isolating the variable.