Comparing Linear Approximations to Calculator Computations
In lecture, we explored linear approximations to common functions at the point x = 0. In this
worked example, we use the approximations to ca
SOLUTIONS TO 18.01 EXERCISES
2. Applications of Dierentiation
2A. Approximation
2A-1
p
d p
b
b
) f (x) a + p x by formula.
a + bx = p
dx
2 a
2 a + bx
By algebra:
2A-2 D(
1
(1
a
p
p
a + bx = a
r
1+
bx
Product of Linear Approximations
Suppose we have two complicated functions and we need an estimate of the value
of their product. We could multiply the functions out and then approximate
the result, o
Compound Interest
If you invest P dollars at the annual interest rate r, then after one year the
interest is I = rP dollars, and the total amount is A = P + I = P (1 + r). This
is simple interest.
For
Hyperbolic Angle Sum Formula
Find sinh(x + y) and cosh(x + y) in terms of sinh x, cosh x, sinh y and cosh y.
Solution
sinh(x + y)
Recall that:
eu + e u
.
2
2
The easiest way to approach this problem m
Solving Equations with e and ln x
We know that the natural log function ln(x) is dened so that if ln(a) = b then
eb = a. The common log function log(x) has the property that if log(c) = d then
10d = c
Evaluating an Interesting Limit
Using lim
n!1
1+
1
n
n
= e, calculate:
3n
1
1. lim 1 +
n!1
n
5n
2
2. lim 1 +
n!1
n
5n
1
3. lim 1 +
n!1
2n
Solution
3n
1
1. lim 1 +
n!1
n
n
1
.
n!1
n
In this problem we