### 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
the answer accords with your physical intuition.
3
```