Concordia MATH 203 Readiness Checklist

1. Simplification and the original domain

Limits often require simplifying an expression near an excluded input. The expression’s domain and its nearby behaviour are different questions. Practise stating the restriction before cancelling.

Common mistake. Cancelling terms before factoring, or forgetting the excluded input.

2. Complete quadratic solutions

A graph’s intercepts and an application’s feasible values often start with an equation. Find every solution first, then decide which ones fit the original context.

Common mistake. Keeping just the positive root without a reason from the problem.

3. Algebraic inputs to a function

Let $x$ be a starting input and let $h$ be the change in that input; the new input is $x+h$. For a function $g$ defined at both inputs, a difference quotient divides the change in output, $g(x+h)-g(x)$, by the nonzero input change $h$. First practise the substitution: replace every occurrence of the input, preserve parentheses, then simplify. In the example the starting input is 2.

Common mistake. Writing $(2+h)^2=4+h^2$ drops the middle term $4h$. Expand both factors before collecting terms.

4. Radians and familiar angles

Review the unit circle, quadrant signs and exact trigonometric values. Radians are especially important for the standard calculus formulas involving sine and cosine. On the circle of radius 1 centred at the origin, cosine is the horizontal coordinate and sine the vertical coordinate. A reference angle is the acute angle between the terminal side and the horizontal axis; use the quadrant to assign the signs.

Common mistake. Using a calculator in degree mode when the input is in radians.

5. Logarithms as inverse operations

Exponential and logarithmic functions appear throughout calculus. First make the exponential the subject of an equation, then apply its inverse. Keep exact values when possible. The natural logarithm $\ln$ has base $e\approx2.718$, so $\ln(e^x)=x$. Its real argument must be positive.

Common mistake. For positive numbers $a$ and $b$, do not split $\ln(a+b)$ into $\ln a+\ln b$. The familiar sum of logarithms comes from a product, not a sum inside the logarithm.

6. Average rate versus average output

A rate measures change in output per unit change in input. A derivative refines this idea locally, so interpret the quotient as well as calculating it. Subtract the initial output from the final output, then divide by final input minus initial input. The two inputs must differ.

Common mistake. Using $\frac{s(3)+s(1)}{2}$ gives the average of the two endpoint positions, not their average rate of change.

Your next revision session

• Mark a topic comfortable only if you can explain every step without a hint.
• Choose two uncertain topics. Study one worked example, solve three new variations, and retry a day later without notes.
• For MATH 203 exam preparation, add derivative-rule selection, curve interpretation and application setup according to your section’s current outline. This checklist alone is not a complete course review.

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