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Marianopolis · Arts & Sciences · 201-AS2-MS

201-AS2-MS Differential Calculus tutoring

Connect limits and derivatives to the quantities they describe. Work through a short motion example, check your algebra and choose what to practise next for Marianopolis Arts & Sciences calculus.

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Your Arts & Sciences calculus course

201-AS2-MS is the Arts & Sciences Differential Calculus course. Marianopolis places Differential Calculus and Mechanics together in semester 1 of its sample program grid. That makes the distinction between position, average velocity and instantaneous velocity a useful connection to practise.

For the Science program, use the separate 201-SN2-RE course entry. Keep the full code from your outline in your lesson request.

Topics to work on

  • Limits, continuity and derivatives from the definition
  • Differentiation techniques, indeterminate forms and L’Hôpital’s rule
  • Graph analysis, optimization and rates of change
  • Linear and higher-order approximations

If L’Hôpital’s rule is among your current topics, try this optional method-choice practice.

Before you start

Review substitution into a function, expansion of a squared bracket and cancellation of a nonzero factor. In the check below, keep the time change nonzero until the difference quotient is simplified.

A velocity is a rate, not a position

An object moves along a line. Its time tt is measured in seconds and its position s(t)s(t) in metres. For 0≤t≤30\le t\le3, let s(t)=1+4t−t2s(t)=1+4t-t^2. Use the derivative definition to find its instantaneous velocity at t=1t=1. Then use that velocity to estimate its position at t=1.1t=1.1, and compare with the exact position.

Show a hint

Let hh be a nonzero change in time from t=1t=1. Form [s(1+h)−s(1)]/h[s(1+h)-s(1)]/h before taking a limit.

Show the reasoning

  1. 1. Identify the starting position

    s(1)=1+4−1=4s(1)=1+4-1=4. The object is at position 4 m when t=1t=1 s. Position is measured in metres; velocity will be measured in metres per second.

  2. 2. Build the average rate of change

    For a nonzero time change hh, with 1+h1+h in the model’s time interval, s(1+h)=1+4(1+h)−(1+h)2=4+2h−h2.\begin{aligned}s(1+h)&=1+4(1+h)\\&\quad-(1+h)^2\\&=4+2h-h^2.\end{aligned}

    The change in position is therefore 2h−h22h-h^2. Dividing it by the elapsed time gives s(1+h)−s(1)h=2h−h2h=2−h.\begin{aligned}\frac{s(1+h)-s(1)}{h}&=\frac{2h-h^2}{h}\\&=2-h.\end{aligned} The cancellation is valid because h≠0h\ne0.

  3. 3. Take the limit

    s′(1)=lim⁡h→0(2−h)=2.s^\prime(1)=\lim_{h\to0}(2-h)=2.

    The instantaneous velocity is 2 m/s in the positive direction. The limit describes the rate at that instant; it does not say the velocity stays constant afterward.

  4. 4. Make the tangent-line estimate

    The time change to t=1.1t=1.1 is 0.1 s. At a rate of 2 m/s, the estimated position change is 2(0.1)=0.22(0.1)=0.2 m. Thus s(1.1)≈s(1)+s′(1)(0.1)=4.2.\begin{aligned}s(1.1)&\approx s(1)\\&\quad+s^\prime(1)(0.1)\\&=4.2.\end{aligned} The estimated position is 4.2 m.

  5. 5. Check the estimate

    Direct substitution gives s(1.1)=1+4(1.1)−(1.1)2=4.19.\begin{aligned}s(1.1)&=1+4(1.1)\\&\quad-(1.1)^2\\&=4.19.\end{aligned} The tangent-line estimate is 0.01 m too high. The term −h2-h^2 explains the difference: the linear estimate kept 4+2h4+2h and left out this quadratic correction.

Your turn

The same model gives s′(t)=4−2ts^\prime(t)=4-2t. At t=2t=2, the instantaneous velocity is zero. Does that mean the position stays unchanged until t=2.1t=2.1? Check using the position formula.

Check your answer

No. s(2)=5s(2)=5 m and s(2.1)=4.99s(2.1)=4.99 m, so the position changes by −0.01-0.01 m. A zero instantaneous velocity at one time does not imply rest over an interval.

If the expansion caused trouble, review the algebra lesson. If the quotient was correct but its meaning was unclear, explain the units of position, displacement and velocity aloud before trying a new function.

What to work through with a tutor

Your tutor can help you choose a first step for a limit or derivative, or review a calculation you have started. Together, you can work through the algebra or derivative rule, then try a new function to check your understanding.

By drawing a graph and working through the calculation with you, your tutor can connect the tangent slope to the rate of change and help you explain the units.

For your optimization or approximation question, your tutor can help define the quantities, build the model with you and check what the answer means.

Make your next practice set count

Choose one limit, one derivative from the definition and one application from your current topics. Write the reason for each step. After checking your work, redo the missed question without looking at its solution.

Independent practice

Continue with these calculus lessons

Official course information

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