Caltech math refresher: solutions to calculus problems Caltech math refresher volunteers September 24, 2014 1 1.1 Calculus Differentiation techniques Example 1. Use the chain rule to evaluate d exp (sin x) . dx Example 2. Use the product rule to evaluate d x3 exp x . dx Example 3. Use the quotient rule to evaluate d x3 + 3 . dx cos x 1.2 Integration techniques Example 1. Use integration by parts to evaluate the following indefinite integral: Z xex dx 1 Example 2. Use the substitution x = cos θ to evaluate the following integral: Z 1 1 √ dx. 1 − x2 −1 Example 3. Use partial fractions to evaluate the following integral: Z 1 dx. 2 x + 3x + 2 1.3 Taylor series Example 1. Find the Taylor series for 1 2 f (x) = √ e−x /2 2π about zero up to O (x5 ). Example 2. Give the Taylor series for ln x about x = 1 up to O (x − 1)3 . 1.4 Partial derivatives Example 1. If f (x, y) = x2 y + exy , calculate Example 2. Let z = x + y and suppose partial derivative of u with respect to x calculate the partial derivative of u with ∂u . Are these two quantities the same? ∂x z ∂f . ∂x that u = z · x. First calculate the ∂u with y held constant ∂x y . Then respect to x with z held constant Why, or why not? Solution 1.5 The multivariable chain rule Example 1. Let u = x2 + xy with x and y. dx dt = 1 and Example 2. Suppose that ∂f =y ∂x and 2 dy dt = 2. Write du dt in terms of ∂f = x + 2. ∂y Use the chain rule to rewrite these equations in terms of the new variables u = x + y and v = x2 . 1.6 Line integrals Example 1. A unit mass moves through a gravitational field. The gravitational field exerts a force F of −10ey N on this mass. The position of the mass at a time t is given by s = (x, y) = (t, 5 − t2 ). Find the work done on the mass by the gravitational field between t = 0 and t = 2, using the ral Z W = F · ds. Example 2. A straight wire stretches between the points (0, 0) and (1, 1). The mass per unit length of the wire is given by ρ (x, y) = yex + x. Find the total mass of the wire. 1.7 Lagrange multipliers Example 1. A positively charged particle is constrained to lie on the circle x2 +y 2 = 1. It reacts to the electric potential generated by another positively charged particle fixed at (2, 2), which is given by φe = 1 q 4π 1 2 . (1) 2 (x − 2) + (y − 2) Use Lagrange multipliers to find the point of lowest potential energy on the unit circle, towards which the constrained particle will travel. Check that the answer accords with your physical intuition. 3