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 calculate values of the sine function near x = 0 and
comp
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 p
bx
a(1 + ), same as above.
a
2a
1
b
1
)=
) f (x)
a
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, or we could approximate each function separately and the
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 compound interest, the year is divided into k equal ti
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 might be to guess that the hyperbolic trig. angle sum fo
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. Its possible to dene a logarithmic function logb (x)
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 do this by using rules of exponents to remove the 3
fro