John Abbott · Science · 201-SN2-RE

201-SN2-RE Differential Calculus tutoring

Get one-to-one help with limits, derivative rules and introductory integration for John Abbott Science 201-SN2-RE. Try the short example below, or send a question or topic you want help with.

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Exam review with explained answers

Your John Abbott Science calculus course

John Abbott’s Science course 201-SN2-RE develops limits, continuity and derivatives, with applications to curves, rates of change and optimization. It also introduces antiderivatives, definite integrals and areas. Your current outline sets the order and depth of these topics.

Use the full code from your outline. This page is for Science 201-SN2-RE; the later Integral Calculus course, 201-SN3-RE, has its own guide below.

Topics to work on

Algebra, functions, limits and continuity
Derivatives of algebraic, trigonometric, exponential and logarithmic functions
Related rates, curve analysis and optimization
Introduction to antiderivatives, definite integrals and areas

A useful starting point

Review factoring, fractions, function notation and exponent rules. For the example, you will use the product rule and the chain rule, then read the sign of a derivative.

Steps worth checking

Keeping both terms in the product rule
Using the derivative’s sign to justify a maximum

Differential calculus: tangent lines and changing quantities

Find a tangent from its slope and a point, then turn a radius rate into an area rate. Keep the units beside each changing quantity.

Before you start: Allow 20–25 minutes to try both questions, then 15 minutes to check and retry any missed steps.

1. From derivative to tangent line

Find the tangent line to $f(x)=\ln(x^2+3)$ at $x=1$ . State its slope and the point used.

Show the explained solution

The chain rule gives $f^\prime(x)=2x/(x^2+3)$ . Hence the slope is $f^\prime(1)=1/2$ .
The point is $(1,\ln4)$ , not $(1,1/2)$ .
$y-\ln4=\frac12(x-1).$ The derivative supplies the slope; the original function supplies the height.

Derivative rules: diagnose the first step

2. Area grows at a changing rate

A circular stain has radius $r(t)$ in centimetres, increasing at $0.40\,\mathrm{cm/min}$ . Find the rate of change of its area when $r=5.0\,\mathrm{cm}$ . Assume the stain remains circular.

Show the explained solution

Write the relationship before substituting the radius: $A(t)=\pi r(t)^2$ .
Differentiate with respect to time: $\frac{dA}{dt}=2\pi r\frac{dr}{dt}.$
At the stated instant, $dA/dt=4\pi\,\mathrm{cm^2/min}$ , about $12.57\,\mathrm{cm^2/min}$ . A constant radius rate does not make the area rate constant.

Use your answers to plan your revision

Slope or value?
Label each number as a function value, a derivative or an input before writing the tangent equation.
Rate with respect to what?
Keep the time-dependent quantities symbolic until after differentiation; check the final units.

Start with a tangent or related-rate question from your allowed review material, even if you have not begun. Your tutor can explain the quantities, help you draw the situation and build the derivative with you. If you have an attempt, you can check its setup together.

Request 201-SN2-RE exam preparation help

Original Nablio practice, not an official examination.

Course topic sources

John Abbott Science: course descriptions

From a derivative to a maximum

Let $x$ be a real variable with $x\ge0$ , and let $f(x)=xe^{-x}$ , where $e$ is the base of the natural exponential. Find $f^{\prime}(x)$ for $x>0$ and use its sign to locate the largest value of $f$ on the given domain.

Show a hint

The expression is a product. Differentiate one factor at a time, then factor out $e^{-x}$ , which is always positive.

Show the reasoning

1. Differentiate the two factors
The derivative of $x$ is 1. The chain rule gives the derivative of $e^{-x}$ as $-e^{-x}$ , because the exponent $-x$ has derivative $-1$ . The product rule therefore gives $\begin{aligned}f^{\prime}(x)&=1\cdot e^{-x}+x(-e^{-x})\\&=e^{-x}(1-x).\end{aligned}$
2. Read the sign
Since $e^{-x}>0$ , only $1-x$ determines the sign. The derivative is positive for $0<x<1$ , zero at 1 and negative for $x>1$ . Thus $f$ increases up to 1 and decreases afterward.
3. State and check the maximum
The function is continuous on $[0,\infty)$ . Its increase followed by decrease makes $x=1$ the global maximum point. The maximum value is $f(1)=e^{-1}=1/e$ . At the endpoint, $f(0)=0<1/e$ . A zero derivative alone would not justify a maximum; the sign change does.

A quick check

On the smaller domain $0\le x\le\tfrac12$ , where does the same function reach its maximum? Do you need a point where the derivative is zero?

Check your answer

The derivative is positive throughout $0<x<\tfrac12$ , so the function increases across this interval. Its maximum is at the endpoint $x=\tfrac12$ , with value $\tfrac12e^{-1/2}$ . An endpoint maximum does not require a zero derivative.

If differentiating was difficult, start with the derivative-rule check. If the calculation was right but the conclusion was unclear, explain what a positive or negative derivative says about the graph.

Next: recover a total from a rate

If your class has reached introductory integration, try the linked lesson’s final exercise. Its cost model uses a linear rate, so the power rule is enough: find an antiderivative, use both bounds for a change, then decide whether a starting value is needed for the total.

Try the rate-to-total exercise This is topic practice in an economic setting. Read its quantity, units and domain before solving; it is not an official John Abbott assessment.

What to work through with a tutor

Bring a limit or derivative question, even if you do not know how to begin. A tutor can help you read the notation, choose a first algebra step or derivative rule, and check the result with you. If you have already started, you can review the first unclear step together.

For an application, your tutor can help define the variable and its domain, build a function from the question and interpret the derivative. If introductory integration is on your outline, ask your tutor to connect an antiderivative with a definite integral and its signed-area interpretation.

Prepare from your own course outline

Choose one limit, one derivative and one application from the topics announced for your assessment. For the limit, check the algebra and the domain; for the derivative, name the rule; for the application, define the variable and explain the result. Redo a missed question without its solution. Include integration only when your teacher has assigned it for this assessment, and bring a question to a review session, with any work you have started.

Independent practice

Choose your next practice step

30 min

Before Calculus: 10 Algebra Skills You Should Know

Start here if fractions, factoring or exponents are interrupting your calculus.

10 min

Derivative Rules: A Three-Question Check

Practise choosing the product, quotient or chain rule before calculating.

18 questions

Calculus I Practice for Limits and Derivatives

Choose questions on limits, continuity and derivatives that match your current topics.

35 min

Sketch a rational function without crossing its domain breaks

Build a three-branch rational-function graph from its domain, one-sided limits and derivative signs. Compare local and global extrema, diagnose false inflection points and try two explained checks.

Course information

Before requesting a lesson

Can I request help with this course?

Send John Abbott 201-SN2-RE, your topic and preferred times. We confirm whether we can help, availability, lesson details and price before payment.

What should I bring?

Bring your current outline and a question you are allowed to share, if you have them. You can show any work you have started or ask your tutor to help you choose the first step.

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Lessons are currently in English, online through Zoom or Google Meet.

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