- Proficient use of the CAS calculator for solving circular function equations, including restricting the domain of the answer. - Similarly proficient use of Desmos Graphing Calculator as a study aid, including restricting domains of functions. -- What you do on Desmos Graphing Calculator can be reproduced on your CAS calculator. - Knew how to solve circular functions graphically.

- Practice writing things down while you still have an incomplete understanding of the problem. The things you write down can be stepping stones that help you understand the problem more completely. - Watch out for clues that can help you solve a given problem. - When you are trying to find the equation of a graphed circular function, start with the basic sin?(x) or cos?(x) or both and introduce transformations step-by-step. Apply dilations before translations. Note that it is possible for a cos?( ) function to be the same as a sin?( ) function with slightly modified translations. The amplitudes and periods will remain be the same. sin?( ) and cos?( ) are out of phase by ?/2 regardless of their period. - Remember to include the relevant units in your answers. - Remember to convert from variable values to meaningful answers depending on the context of the problem. - It is possible that you may be finding doing algebraic working out easier when you write x instead of any other variable such as t. Practice doing working out using the other variables while still imagining that those variables are like x. The reason for this is to avoid confusing people who read your working out when they expect to see t but find x, not being certain what you mean by x.

- Calculated the derivative of f(x)=?(1-2x) correctly using the chain rule. You differentiated the expression in one go without explicitly writing the implied variable substitution. - Successfully found the angle from the positive direction of the x-axis to the tangent to a curve at a given x value, measured in the anticlockwise direction. - Correctly interpreted the meaning of f(g(x)) given f(x) and g(x). - Correctly understood that finding the inverse of a function means swapping x and y. Successfully solved for y in the inverse equation. Also successfully constrained the domain of the inverse equation given the constrained domain of the original equation, albeit with a hint from the tutor. (Q5(b) in 2016 MM Exam 1)

Domains and Ranges of Graphs - You can make some intermediate steps towards finding the domain and range of a given function. - Practice checking how the restricted domain of a function affects the domain of the inverse of the function. Substitutions like h(-x) - What does h(-x) mean? What does h(x) mean? What is the difference between them? h(x)=log_e?(x^2+1). h(a)=log_e?(a^2+1). h(-x)=log_e?((-x)^2+1). - In all cases, we replaced the variable x with whatever was in the brackets of h, including – signs. Notation - Remember that dy/dx is a term whereas d/dx is an operator. - Be more careful when substituting a variable.

- Proficiently used the slope-point form of linear equations to determine the equation of a tangent line. (Q2(c)(ii) in 2016 MM Exam 2) - Correctly identified that the y value of the point located at the intersection between two lines is found by equating the equations of the lines to each other, i.e., y_1 (x) = y_2 (x), where the lines have equations of the form y = y_1 (x) and y = y_2 (x). (Q2(c)(ii) in 2016 MM Exam 2) - Successfully implemented Pythagoras Theorem to calculate the distance between two points. (Q2(c)(ii) in 2016 MM Exam 2)

Connections between ideas - In Q2(c)(i) in 2016 MM Exam 2, the goal was to determine the coordinates of point D. You were given the full equation of the tangent at point A and told that the tangent at point D is parallel to the tangent at point A. What does this mean? The tangent at D has the same gradient as the tangent at A. And what has the same gradient as a tangent? The derivative of the curve at the point where the tangent touches. Now you have a derivative equation to solve for the x value of the coordinates of D. Connect new concepts to familiar concepts - You worked with simultaneous equations before. A matrix equation is a set of simultaneous equations. (Q1(e) in 2016 MM Exam 1) Breaking down a problem into a series of steps - See f(x) or f'(x)? Replace them with their actual expressions. - Don't know how to proceed? Check what info you have and what you can do with it. Mathematical concepts - Flip the inequality sign when reciprocating both sides of the inequation or multiplying or dividing the sides by a negative number. - By definition, the area to the right or left of the median of a probability density function is 0.5. What can you do with this info? - Some circular functions have pi in their coefficient for x. Remember to include the pi in the n for 2*pi/n for sin and cos and in pi/n for tan when calculating periods. - Discrete probability distribution functions. Understanding/Decoding questions by breaking them down - Identify nouns in the question. Nouns are things you can find or work with to solve the question. Eliminating choices in multiple-choice questions - Look at your options. Look at how they differ from each other. What info are you given in the question that can help you eliminate some of the options? What info gives you a hint for which option is the correct one?

- Graphed f(x) when you wanted to determine its range. (Q1 in 2016 MM Exam 2) - Correctly identified that the minimum y value of y = 2 cos?(x/2) + ? occurs at a half-period away from where the maximum y value occurs along the x-axis. (Q1 in 2016 MM Exam 2) - Getting the hang of writing out on the exam/test paper the steps you take on your CAS calculator. (Q2 in 2016 MM Exam 2) - Remembered that the gradient of the line that is perpendicular to the tangent to the curve f(x) at x is -1/m, where m is the gradient of the tangent itself. (Q2 in 2016 MM Exam 2) - Successfully found the tangent line to a curve g(x) at x. - Successfully found some key points on the tangent lines. - Successfully differentiated an expression with an unknown constant to show that the given expression is an antiderivative of another given expression. (Q8 in 2016 MM Exam 1) - Successfully calculated a probability Pr?(X>x) for a continuous random variable using integration.

Review some concepts - Binomial distribution. Signs for when to use a binomial distribution. How to turn these signs into a random variable. - Revise how to find the equation of the tangent to a curve f(x) at x. Working out - Remember to write all brackets. - You extracted the triangle in Q2 in 2016 MM Exam 2 to calculate its area. You drew a version of the triangle that was rotated by 90 deg. You correctly carried over the base length of the triangle but incorrectly carried over its height, saying that the height was 1/3 units. You soon corrected this mistake with a bit of help from the tutor, correctly saying that the height was 2 units. Please consider drawing rotated axes to help you keep track of how the rotation affects lengths and heights. The tutor drew such axes in green pen next to the triangle you drew for Q2 as an example. - Remember to explicitly write “? Shown” after you have written your working out for showing something. That way, you clearly tell the person looking at your work that you are sure you have shown what was requested.