### Caltech math refresher: solutions to calculus problems

```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