Western Calculus 1501A/B Calculus II: Topics, Guide and Practice

Calculus 1501A/B is Calculus II for Mathematical and Physical Sciences at Western University. The calendar says students who intend to pursue a degree in actuarial science, applied mathematics, astronomy, mathematics, physics or statistics should take it. It is a half course with four lecture hours, and it combines integration techniques, a careful look at the Mean Value Theorem, series, polar curves and first order differential equations. This guide maps each topic to free practice.

What Western Calculus 1501A/B covers

The Western academic calendar lists the topics as: techniques of integration; the Mean Value Theorem and its consequences; series, Taylor series with applications; parametric and polar curves with applications; and first order linear and separable differential equations with applications. The prerequisite is a minimum mark of 60% in Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B or the former Applied Mathematics 1412A/B.

This is a full Calculus 2 course. The Mean Value Theorem section is what sets it apart from Calculus 1301A/B: it is the tool used to prove many results in calculus, which matters for students heading into mathematical degrees.

Calculus 1501A/B topic map

Calculus 1501 vs Calculus 1301

They are antirequisites. Before this course you take Calculus 1000A/B or 1500A/B, and after it Calculus 2302A/B.

Example problems at the Calculus 1501 level

1. A consequence of the MVT. If f'(x) = 0 for every x in an interval, then f is constant there: for any a < b, f(b) - f(a) = f'(c)(b - a) = 0.

2. A first order linear equation. Solve y' + y = e^{x}. The integrating factor is e^x, so (e^x y)' = e^{2x} and

3. A polar slope. For r = \theta, x = \theta\cos\theta and y = \theta\sin\theta, so \frac{dy}{dx} = \frac{\sin\theta + \theta\cos\theta}{\cos\theta - \theta\sin\theta}.

4. Taylor series. \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots, so \int_0^1 \cos(x^2)\,dx \approx 1 - \frac{1}{10} + \frac{1}{216} \approx 0.905.

How to do well in Calculus 1501A/B

• Practise integration until it is quick. Every later topic, from polar area to ODEs, ends in an integral.
• Learn the MVT with its hypotheses. Expect questions that ask you to use it to prove an inequality or show a function is constant.
• Know the core series. Geometric series, p-series and the Maclaurin series for e^x, \sin x and \cos x come up constantly.
• Classify each ODE first. Decide whether it is separable or first order linear before you start solving.

Tip: for a first order linear equation y' + P(x)y = Q(x), the integrating factor is e^{\int P(x)\,dx}. Multiply through and the left side becomes one derivative.