a calculate f0 Hint use the trig identity involving arctan a

a) calculate f(0). Hint: use the trig identity involving arctan and tan

b) find the limit expression for f\'(0)

d) solve for f(x) by soliving the differential equation

Solution

We introduce the vector space V spanned by the vector 0=(a,b):

We say that 0 is a basis vector in the space V. Our aim is to find the vector uu=c00V which best approximates the given vector ff=(3,5). A reasonable criterion for a best approximation could be to minimize the length of the difference between the approximate uu and the given ff. The difference, or error, ee=ffuu has its length given by the norm

where (ee,ee) is the inner product of ee and itself. The inner product, also called scalar product or dot product, of two vectors uu=(u0,u1) and vv=(v0,v1) is defined as

Remark 1. We should point out that we use the notation (,) for two different things: (a,b) for scalar quantities a and b means the vector starting in the origin and ending in the point (a,b), while (uu,vv) with vectors uu and vv means the inner product of these vectors. Since vectors are here written in boldface font there should be no confusion. Note that the norm associated with this inner product is the usual Eucledian length of a vector.