This is a minimal set of objectives. You are also expected to solve problems that combine different skills and demonstrate ability to apply the learned concepts in situations you haven't seen before.
Lines and distances
1. Write down the equation of a line in the form y - y0 = m(x - x0) given two points, a point and the
slope, or a graph.
2. Sketch a line given its equation in any form (e.g. x/3 + 19 + 5y = -7).
3. Determine if lines are perpendicular or parallel.
4. Explain, using a picture, the Pythagorean Theorem.
5. Find the distance between two points on the plane using the Pythagorean Theorem.
6. Write down the equation of a circle in the form (x - x0)2 + (y - y0)2 = r2 given its radius and centre.
7. Sketch a circle given its equation in the form (x - x0)2 + (y - y0)2 = r2.
1. Explain what a function is.
2. Give examples of functions and their domains, including functions with restricted or unusual domains
(e.g. all positive numbers with the exceptions of 1, 2, 3).
3. Calculate the domain of functions of the form p(x)/q(x) where p and q are compositions of polynomials and
4. Sketch the functions x, x2,x3, sqrt(x), |x|, 1/x
5. Sketch the graphs of functions obtained from x, x2,x3, sqrt(x), |x|, 1/x via one or more of the following
transformations: vertical and horizontal shifts, vertical and horizontal dilations, and reflections across
the vertical and horizontal axes.
6. Explain, using a picture or examples, what the composition of two or more functions means.
7. Explain, using an example, why composition of f and g is not the same as composition of g and f.
Velocity and the idea of limit
1. Explain, using a picture but not the formal vocabulary of limits, what velocity and acceleration are.
2. Sketch the graphs of velocity and acceleration of a particle given a graph of its position.
3. Given graphs of the position, velocity and acceleration of a particle, identify which is which.
4. Numerically estimate the velocity of a particle given a graph of its position.
5. Explain, using a picture, what a di.fference quotient is.
1. Explain, using a picture and in words, what the phrases "limit as x->a of f(x) is L", "limit as x->a- of f(x) is
L" and "limit as x->a+ of f(x) is L" mean.
2. Argue why the notion of limit is important.
3. Calculate the limits of simple rational functions whose limits exist (e.g. limit as x->2 of (x - 2)/(x2 - 5x + 6) ).
4. Give examples of rational functions whose limits at a given .finite number of points do not exist.
1. Define what it means for a function to be continuous.
2. Determine the parameters of a piecewise function that make a function continuous or di.fferentiable.
3. State and explain the Intermediate Value Theorem.
4. Give examples of functions that do not satisfy the Intermediate Value Theorem by virtue of their
discontinuity either in the interior of an interval or at its endpoints.
5. Use the Intermediate Value Theorem to estimate roots of functions, including non-polynomial functions.
6. Use the Intermediate Value Theorem to construct short proofs (e.g. "prove that a function f : [0, 1] ->
[0, 1] has a fixed point").
Velocity and the idea of derivative
1. Identify that the slope of a tangent line is a limit.
2. Estimate slopes of tangent lines using the definition of derivative.
3. Explain why the velocity is the derivative of position as a function of time and the acceleration is the derivative of velocity as a function of time.
The definition of derivative
1. Define the derivative of a given function using limits.
2. Calculate derivatives, using the limit definition, of polynomial functions of degree at most 3, rational
functions p(x)/q(x) where p and q are linear, sqrt(x), functions with piecewise definition.
3. Find the equation of lines tangent to the graphs of functions like in 1.
---Midterm 1 covered objectives up to this point---
4. Define what it means for a function to be differentiable.
Given a function definition with parameters, identify the conditions on the parameters that make the function differentiable.
5. Explain why a differentiable function is continuous, but a continuous function might fail to be differentiable.
6. Explain why (f+g)'=f'+g', (f-g)'=f'-g', (cf)'=c f' for a constant number c.
Power, product and quotient rules
1. Explain why (a+b)n=an+n an-1b + ... where the omitted terms are of the form (an integer times an-kbk) for k>1.
2. Show using the expansion of (a+b)n above and the definition of derivative why (xn)'=nxn-1
3. Apply the power rule (xn)'=nxn-1 with exponent n that is not necessarily a positive integer to compute (1/x)', (sqrt(x))' etc.
4. Explain why the product rule works the way it does.
5. Explain how the quotient rule can be derived from the product rule.
6. Apply the product, quotient and power rules to differentiate functions.
1. Explain what a radian is.
2. Convert angle measurements from degrees to radians and back.
3. Define sin x and cos x in terms of the horizontal and vertical coordinates of a point on the unit circle.
4. Identify sin x and cos x as ratios of sides in a right angled triangle with one of the angles equal to x.
5. Sketch the graphs of sin x and cos x
6. Compute sin x and cos x for some common angles (k π/2 ± x for x=0, π/6, π/3, π/4, π/2 and integer k).
7. Explain what the derivatives of sin x and cos x are.
0. Do arithmetic with exponentials (e.g. simplify 4324/82/3)
1. Explain what a differential equation is.
2. Explain why f(x)=ex satisfies the differential equation f'(x)=f(x)
3. Know that the solutions of the differential equation f'(x)=rf(x) are of the form f(x)=cerx
4. Model a word problem involving the rate of growth of a function using a differential equation.
5. Sketch the graphs of functions erx for different values of r.
6. Define and sketch the function ln x
7. Apply algebraic identities to simplify expressions with ln
8. Differentiate functions that involve ln x and erx
The chain rule
1. Explain why the chain rule works the way it does
2. Differentiate functions using the chain rule
3. Find tangent lines, normal lines and tangent lines with given slope for the graph of a function y=f(x), where f(x) is any function for which you can write an algebraic expression.
1. Find y' for the function y(x) satisfying an equation of the form f(x,y)=g(x,y)
2. Apply logarithmic differentiation to find y' for y(x) that is a function with definition involving powers, products and quotients
---December final covers objectives up to this point---