Intro to Calculus

What this covers

• Evaluating limits algebraically (factoring, rationalising, one-sided limits) and reading limits from graphs, including where a limit exists but the function value doesn't
• Defining the derivative as a limit of difference quotients, then moving to the power, product, quotient and chain rules
• Interpreting f'(x) as slope of a tangent line and as a rate of change, including velocity and acceleration from a position function
• Using the first and second derivatives to find critical points, intervals of increase/decrease, concavity and inflection points, and to sketch curves
• Setting up and solving optimisation and related-rates word problems, including choosing variables and writing the constraint equation
• Antiderivatives, the definite integral as accumulated area, and using the Fundamental Theorem of Calculus to evaluate simple definite integrals

Where learners get stuck

Believing the limit of f(x) as x approaches a is the same thing as f(a)

For every polynomial they meet first, substitution just works, so students generalise the shortcut. Then they hit 0/0 forms and holes in graphs and have no separate mental model for 'what the function approaches' versus 'what the function equals'.

Applying the chain rule inconsistently — differentiating the outer function but forgetting the inner derivative, especially inside trig or exponential functions

With simple cases like (3x)^2 the missing factor is easy to absorb into an answer that still looks plausible. Students learn the rules as a list of surface patterns rather than as composition, so they don't ask 'what is the inside function here?' before starting.

Treating a derivative of zero as automatically meaning a maximum or minimum

Textbook examples are mostly well-behaved parabolas and cubics. Students memorise 'set the derivative to zero' as the whole method and skip the sign test or second-derivative check, so saddle-type points and endpoint maxima get missed in optimisation problems.

What a session looks like

A session usually starts with you describing a problem you're stuck on or a topic from class, then working through it out loud while Evelyn Tutor prompts for the next step rather than supplying it. Expect to be asked to justify a step — why factoring is legitimate here, what the derivative you just found actually measures, whether your optimisation answer is a maximum. Written work happens on your own paper or screen and you describe it; graphs and limit behaviour are talked through verbally, which is good practice for explaining calculus reasoning in exam write-ups.

Helpful to know first

• Fluent algebra: factoring quadratics and cubics, simplifying rational expressions, and rationalising expressions with radicals
• Function notation, domain and range, and composition of functions
• Graph behaviour of polynomial, rational, exponential, logarithmic and trigonometric functions, including asymptotes
• Slope of a line and the equation of a line through a point, since tangent lines depend on both

Questions

What's the difference between Intro to Calculus and Pre-Calculus?

Pre-Calculus builds the function toolkit — transformations, trig identities, exponentials, sometimes an early look at limits. Intro to Calculus uses that toolkit to study rates of change and accumulation: derivatives, tangent lines, optimisation and the definite integral. If a student is still unsure about function composition or trig graphs, that Pre-Calculus gap will surface quickly in the chain rule and in derivatives of sin and cos.

My child can do the derivative rules but fails word problems. What helps?

Related-rates and optimisation problems fail at the setup stage, not the calculus. Sessions target that specifically: naming the quantities, writing the relationship between them before differentiating, and deciding what the question is actually asking for. Talking the setup through aloud tends to expose the missing step faster than reworking more mechanical derivative drills.

Do you cover integration, or just derivatives?

Both, at an introductory level: antiderivatives, basic substitution, the definite integral as area under a curve, and the Fundamental Theorem of Calculus linking the two. Techniques like integration by parts and partial fractions belong to a later course and aren't the focus here.

How much do we need to know about limits before derivatives make sense?

Enough to evaluate a 0/0 limit by factoring or rationalising, to handle one-sided limits, and to read continuity off a graph. The derivative is defined as a limit of a difference quotient, so students who skip limits tend to memorise the power rule with no idea where it came from — which makes later topics like implicit differentiation harder to hold together.