McGill University · MATH 262

McGill MATH 262: intermediate calculus tutoring

Work through Taylor approximations and multivariable calculus in MATH 262. A tutor can help you identify the point of approximation, justify an error bound or sketch the region before setting up an integral. Start with a question or a concept; we will confirm support for your topic before booking.

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Online 60 min: $55 CAD · Pricing Midterm & final exam help Check a Taylor approximation and its error

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Course overview

MATH 262 is Intermediate Calculus for Engineering students. It connects Taylor series with vector geometry, multivariable differentiation, optimization and multiple integration. Send the chapter or a question you want help with so we can confirm support before booking. If you have an attempt, you can include it.

Open only to students in the Faculty of Engineering.

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Build a Taylor estimate and justify its accuracy
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MATH 262 topics to review

Series, power series and Taylor’s theoremVector geometry, vector functions and curvesMultivariable derivatives and approximationConstrained and unconstrained extremaMultiple integrals, surface area and change of variables

Common difficulties

Visualizing surfaces and regions
Selecting coordinates and bounds
Knowing when a theorem or method applies

A Taylor value needs an error check

Let $x$ be dimensionless and $f(x)=1/(1+x)$ , defined for $x\ne-1$ . Use its order-two Taylor polynomial $P_2$ centred at zero to approximate $f(0.1)$ . Can you certify an absolute error at most $0.001$ ? Name the whole interval on which your derivative bound must hold.

Hint: bound the derivative between centre and target

For an order-two Taylor polynomial, use a bound $M$ for $|f^{(3)}(t)|$ on the segment from zero to the target. Then $|f(x)-P_2(x)|\le M|x|^3/3!$ . The segment must stay inside the function’s smooth domain.

Build the polynomial and certify the estimate

The values $f(0)=1$ , $f'(0)=-1$ , $f''(0)=2$ give $P_2(x)=1-x+x^2$ , so $P_2(0.1)=0.91$ . The relevant segment is $[0,0.1]$ , where $f^{(3)}(t)=-6/(1+t)^4$ has magnitude at most $6$ . Therefore

|f(0.1)-0.91|\le\frac{6(0.1)^3}{6}=0.001.

The bound certifies the requested tolerance. The exact comparison is $|10/11-91/100|=1/1100<0.001$ . The decimal $0.91$ is the exact polynomial value, so no rounding allowance is needed here.

Retry: move to the other side of the centre

Use the same polynomial at $x=-0.1$ . Find a valid $M$ on the new segment. Does the resulting bound certify error at most $0.001$ ? Then compare with the exact value to see whether the tolerance itself is met.

Check the interval and the actual error

Now $P_2(-0.1)=1.11$ and the segment is $[-0.1,0]$ . The largest derivative magnitude occurs at $t=-0.1$ , giving $M=6/(0.9)^4$ . Thus

|f(-0.1)-1.11|\le\frac{(0.1)^3}{(0.9)^4}=\frac{10}{6561}\approx0.001524.

This bound does not certify the target. A failed bound alone would not prove that the actual error is too large. Here the exact comparison does: $|10/9-111/100|=1/900>0.001$ . Reusing the centre value $M=6$ would be unjustified because the derivative magnitude increases to the left.

Continue with the existing four Taylor-error questions. If your current obstacle is multivariable integration, use the region-and-bounds practice instead. These are topic routes within the broader course.

How tutoring can help

You can bring a Taylor-approximation question even if you have not started it. A tutor can help you build the polynomial, identify the target input and find an error bound over the whole interval. If you already have an attempt, you can check the derivative order and whether the bound meets the requested tolerance together. Then move the target across the centre and justify a new bound.

Exam preparation for MATH 262

For Taylor questions on your exam list, justify the derivative bound over the whole interval and decide whether it certifies the required accuracy. Retry the negative-input example without the solution. For multiple integrals, check the region and bounds; for optimization, justify the classification of each stationary point.

Use your section’s outline to choose the chapters to review. We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.

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