Question

Problem A.1

The graph below is made of three line segments:

-1 1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

4

y

x

f(x)

g(x)

h(x)

The segments correspond to the following three functions:

f(x) = x − 2, g(x) = p

4 − (x − 6)2 + 2, h(x) = x − 6

Find the total length L of the graph between x = 2 and x = 10.

Problem A.2

Let f(x), g(x) and h(x) be the functions from Problem A.1. Find the derivative λ

(x) of the

following function with respect to x:

λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x)

*Please give specific answers to both Problem A.1 & A.2

Answer 1

A 1

$f(x) = x − 2$ for $x\isin [2,4]$

$g(x) = \sqrt{4 − (x − 6)^2} + 2$ for $x\isin [4,8]$

$h(x) = x − 6$ for $x\isin [8,10]$

then:

length of f(x):

$L_1=\sqrt{2^2+2^2}=2\sqrt 2$

length of h(x):

$L_2=\sqrt{2^2+2^2}=2\sqrt 2$

length of g(x):

$L_3=2\pi R/2=2\pi$

total length:

$L=L_1+L_2+L_3=2\sqrt 2+2\sqrt 2+2\pi=2(\pi+2\sqrt 2)$

A 2

$f(x) · g(x)=(x-2)(\sqrt{4 − (x − 6)^2} + 2)$

$f(x) · h(x)=(x-2)(x-6)$

$g(x) · h(x)=(x-6)(\sqrt{4 − (x − 6)^2} + 2)$

$λ(x)=(x-2)(\sqrt{4 − (x − 6)^2} + 2)+(x-2)(x-6)-(x-6)(\sqrt{4 − (x − 6)^2} + 2)$

$λ'(x)=\sqrt{4 − (x − 6)^2} + 2-\frac{(x-2)(x-6)}{\sqrt{4 − (x − 6)^2} }+2x-8+\sqrt{4 − (x − 6)^2} + 2-\frac{(x-6)^2}{\sqrt{4 − (x − 6)^2} }=$

$=2\sqrt{4 − (x − 6)^2}+2x-4-\frac{(2x-8)(x-6)}{\sqrt{4 − (x − 6)^2} }$