U of T MAT136H1 Calculus II: Topics, Study Guide and Practice

MAT136H1 is Calculus II at the University of Toronto and the second course in the default calculus sequence for science students, right after MAT135H1. Where MAT135H1 is about derivatives, MAT136H1 is about integrals and what you can do with them, from areas and totals to infinite series and differential equations. This guide maps every MAT136H1 topic to practice so you can learn by solving problems.

What U of T MAT136H1 covers

According to the Arts and Science calendar, MAT136H1 opens with the Fundamental Theorem of Calculus, which ties accumulating change to finding area. From there students use integrals to compute areas, volumes and totals, to analyze equations involving derivatives, to make sense of infinite sums, and to write complicated functions as "infinite polynomials".

The listed topics are integration by substitution and by parts, improper integrals, computer algebra systems for integration, Taylor polynomials and Taylor series, the ratio test and radius of convergence, and first-order differential equations and systems, including separable equations and modelling. That matches the integral half of a standard Calculus 1 course plus the core of Calculus 2, without trig substitution, partial fractions, parametric curves or polar coordinates in the listed topics.

MAT136H1 topic map

MAT136H1 vs MAT149H1

Both count as a second calculus course for programs that ask for MAT135H1 and MAT136H1. If you later want proof-based multivariable calculus in MAT237Y1, the calendar pairs MAT135H1 and MAT136H1 with a proofs course such as MAT138H1, plus linear algebra.

Example problems at the MAT136H1 level

1. Integration by parts. Evaluate \int x\cos x\,dx with u = x, dv = \cos x\,dx:

2. A Taylor polynomial. The degree 3 Taylor polynomial of e^x at 0 is

so e^{0.1} \approx 1 + 0.1 + 0.005 + 0.000167 \approx 1.10517.

3. A separable differential equation. Solve \dfrac{dy}{dt} = 2ty with y(0) = 3. Separate and integrate:

so y = Ae^{t^2}, and y(0) = 3 gives y = 3e^{t^2}.

How to do well in MAT136H1

• Learn to pick the method. Most integration mistakes come from choosing the wrong technique. Our choosing a strategy lesson trains exactly that decision.
• Treat series as pattern recognition. Memorize the common Maclaurin series and the ratio test, then read the convergence tests guide.
• Use the formula sheet. The calculus formula sheet lists the integrals and series you will use most.
• Practice under time pressure. Mixed sets in practice with fresh numbers each time get you ready for term tests, and how to prepare for a calculus exam has a study plan.

Tip: check every antiderivative by differentiating it. It takes seconds and catches most sign errors before they cost marks.