Exam Session: How Do You Solve a Problem?

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This session is less about new content and more about how you attack a problem you haven’t seen before.

I. Problem-solving strategies

In bullet points, describe your problem-solving strategy.
What do you do when you don’t know where to start on a question?
Imagine the ideal problem-solving strategy and write it down in bullets. How does it differ from your answers to 1 and 2?

Here is some of Terence Tao’s advice on solving mathematical problems (from Solving Mathematical Problems: A Personal Perspective):

Understand the problem.
Understand the data.
Understand the objective.
Select good notation.
Write everything down.
Modify the problem slightly.
Simplify, exploit the data, and reach tactical goals.

For each point, give one example you’ve already run into in this course.

II. Try it on a problem

The line from \((-1,0)\) through a point \((x,y)\) on the unit circle crosses the \(y\)-axis at \((0,t)\).

a. Express \(\alpha\) in terms of \(\theta\). What geometric property justifies this relationship?
b. What is the slope of the line from \((-1, 0)\) to \((x, y)\) in terms of \(t\)? And in terms of \(x\) and \(y\)?
c. Using \(x^2+y^2=1\) and the slope from part b, express \(\sin\theta\), \(\cos\theta\) and \(\tan\alpha\) in terms of \(t\).
d. Using implicit differentiation, find \(d\theta\) in terms of \(t\).
e. Use parts c and d to evaluate \(\displaystyle \int \frac{d\theta}{\sin^2\theta+\cos\theta+2}.\)

Reflection

Go back through the problem and write down, in bullets, the method you actually used. Where did it match your ideal strategy? How could you improve your technique?