All the skills below should be trained by solving problems, writing down their solutions, evaluating the solutions with the help of TA/teacher or by comparing to available solutions, and reflecting on the weaknesses identified in the evaluation. Understand what a given problem asks for Identify strategies and techniques that might help with solving the problem Formulate a strategy that will lead to the solution of the problem Follow this strategy performing necessary calculations along the way Adjust the strategy based on new information discovered while trying to implement it Write up the obtained solution in a way that makes it readable Critically evaluate the written solution Skills: Find the equation of a line given two points on the line/point on a line + its slope Given an equation of a line identify its slope Find the slope of a line parallel/perpendicular to a given one Find the equation of a circle given its radius and center Given an equation of a circle identify its radius and center Identify a given equation as the equation of a line or a circle. Sketch the corresponding geometrical shape. Find points of intersection of two lines/a line and a circle/two circles Solve a quadratic equation Find the distance between two points Find the length of the third side in a right-angled triangle with given lengths of the other two sides. Find the lengths of all the sides and magnitudes of all the angles in a right-angled triangle with given length of one of the sides and given magnitude of one of the angles that is not the right angle. Given a function f having a relatively simple algebraic expression that involves sums/differences, powers of x, quotients, products, trigonometric functions, logarithms, exponentiation and possibly a definition by cases,
Find an algebraic expression for the composition of two or more functions Find the average rate of change/slope of the secant line of a function on an interval Find the instantaneous rate of change/slope of the tangent line of a function at a point Find the derivatives of simple functions using the definition of the derivative Find the derivative of a complicated function using constant multiple, sum, difference, product and quotient rules, the chain rule, logarithmic differentiation and the derivatives of the functions x ^{n}, sin x, cos x, ln x, e^{x}Identify instantaneous velocity as the derivative of position with respect to time and instantaneous acceleration as the derivative of the velocity with respect to time Evaluate (possibly one-sided) limits of functions Sketch/find an algebraic expression for a function that satisfies some given properties Show that there is at least one/several solutions of a given equation on a given interval using Intermediate Value Theorem or using some algebraic technique/guessing a solution. Find slope of the tangent line to a curve given by an equation of the form f(x,y)=g(x,y) at a point lying on the curve. Simplify expressions that involve exponents and logarithms Evaluate trigonometric functions at some values of the argument Solve simple trigonometric equations (e.g. sin x = 1/2) Work with angles measured in radians Apply the chain rule in situations where the expression to be differentiated involves unknown functions Know when and how to apply an exponential growth model Determine whether a statement is true or false. Prove it if it is true or find a counterexample if it is false (the statement is some statement about lines, circles, functions, limits, continuity, differentiability, derivatives, existence of solutions to some equations, differentiation rules, trigonometric functions, exponentiation, logarithms or curves defined by an equation). |