Final Examination

At a glance

Individual · three hours, in the examination period · written, with a calculator and one formula sheet supplied · both halves of the course, weighted as the course spent its time

What it is for

Everything else this semester was worked in groups, at the boards, with time and somebody beside you. This is the one piece of evidence that is unambiguously yours — and, more usefully, it is the closest thing you will meet to a first-year university mathematics assessment while somebody who knows you is still the one marking it.

What is on it

Part	Roughly	What it asks you to do
A. Limits and the derivative	20%	Evaluate limits; differentiate from first principles; interpret rates of change numerically and graphically
B. Differentiation and its uses	30%	Product, quotient, and chain rules; implicit and higher-order derivatives; exponential, logarithmic, and sinusoidal derivatives
C. Curve sketching and optimisation	20%	Sketch $f^{\prime }$ from $f$; use the second derivative; full curve analysis; solve an optimisation problem from a described situation
D. Vectors, lines, and planes	30%	Vector operations, dot and cross products, projections; equations of lines and planes; intersections and what each case means geometrically

What to expect, precisely

• One first-principles question. The rules are not permitted for it; the limit definition is the answer being marked. Derivatives from First Principles is the method, and the marks are in the cancellation step.
• “Sketch $f^{\prime }$ given $f$” appears every year. You are reading slopes off a graph and plotting them — no equation is given, and none is needed. The second derivative question is the same skill applied twice: it asks where the slope’s own rate of change turns.
• Optimisation is set in words, not handed to you as a function. The marks are distributed across defining the variable, building the model, differentiating, testing the critical point, and answering the question that was actually asked — including units.
• The vectors half is not an afterthought. It is thirty per cent, it is the part students under-prepare, and the intersection questions reward saying what the answer MEANS: a point, a line, no intersection, or coincident planes.
• Exact values where they exist. $\frac{\pi }{3}$ is an answer; 1.047 is a rounding of one.

How to prepare

1. Redo problems, do not reread them. Five from each unit’s practice set, cold, with only the formula sheet: Limits Practice, Derivative Rules Practice, Chain Rule Practice, Curve Sketching Practice, Optimization Practice, Dot and Cross Product Practice, Lines and Planes Practice.
2. Rebuild the connections. A derivative is a slope is a rate of change is a velocity — one idea in four vocabularies. Draw that map from memory; the gaps are your study list.
3. Re-derive one rule. The product rule from first principles, once, with the paper closed afterwards. If you can do it, you understand what a derivative is; if you cannot, that is where to spend an hour.
4. Practise reading graphs of $f^{\prime }$ and $f^{\prime \prime }$ — the most common place marks are lost, because it is the least practised.
5. Bring questions to the review classes. This page is what is on the examination.

In the three hours

The vectors half is more mechanical and less error-prone under pressure than the calculus half. Doing it second, while tired, is the usual mistake — it is the part most likely to be finished quickly and correctly if you are alert. If a question stalls, write the method you intended in words and move on; a described method earns marks and a blank space earns none.

How this is assessed

Against the same expectations as everything else. Per How Marks Work, this examination is part of the final 30% of the course mark alongside The Math Symposium, so that neither one afternoon nor one project decides your grade alone.

Curriculum connection

A2.3

determine the derivatives of polynomial functions by simplifying the algebraic expression $\frac{f(x+h)-f(x)}{h}$ and then taking the limit of the simplified expression as $h$ approaches zero [i.e., determining ${\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$]

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B1.1

sketch the graph of a derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function (i.e., points at which the concavity changes) Sample problem: Investigate the effect on the graph of the derivative of applying vertical and horizontal translations to the graph of a given function.

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B1.2

recognize the second derivative as the rate of change of the rate of change (i.e., the rate of change of the slope of the tangent), and sketch the graphs of the first and second derivatives, given the graph of a smooth function

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C1.1

recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors (e.g., displacement, forces involved in structural design, simple animation of computer graphics, velocity determined using GPS) Sample problem: Position is represented using vectors. Explain why knowing that someone is 69 km from Lindsay, Ontario, is not sufficient to identify their exact position.

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C1.2

represent a vector in two-space geometrically as a directed line segment, with directions expressed in different ways (e.g., $320^{\circ }$; N 40° W), and algebraically (e.g., using Cartesian coordinates; using polar coordinates), and recognize vectors with the same magnitude and direction but different positions as equal vectors

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C3.1

recognize that the solution points $(x,y)$ in two-space of a single linear equation in two variables form a line and that the solution points $(x,y)$ in two-space of a system of two linear equations in two variables determine the point of intersection of two lines, if the lines are not coincident or parallel Sample problem: Describe algebraically the situations in two-space in which the solution points $(x,y)$ of a system of two linear equations in two variables do not determine a point.

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C3.2

determine, through investigation with technology (i.e., 3-D graphing software) and without technology, that the solution points $(x,y,z)$ in three-space of a single linear equation in three variables form a plane and that the solution points $(x,y,z)$ in three-space of a system of two linear equations in three variables form the line of intersection of two planes, if the planes are not coincident or parallel Sample problem: Use spatial reasoning to compare the shapes of the solutions in three-space with the shapes of the solutions in two-space for each of the linear equations $x=0$, $y=0$, and $y=x$. For each of the equations $z=5$, $y-z=3$, and $x+z=1$, describe the shape of the solution points $(x,y,z)$ in three-space. Verify the shapes of the solutions in three-space using technology.

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