Help for exam(differential calc)

[matrix_204]

I needed some help on solving some problems that we haven't really covered in class but are expected to know. For example i was looking at the exam written from previous years and it had some questions that i wasn't able to do it right away. I need explaination for the following,

1. If there is a question asking to find a nonconstant polynomial function, ie., lim f(x)/g(x)=13 or 0 or d.n.e. (as x=>2). How do i find these functions?

2. If a function is defined by a formula with given constants a,b,c,d. How do i find a formula for the inverse of the function with x (element of) range(f)? E.x., f(x)=(ax +b)/(c +dx).

3. How do I show that an unknown function satisfies some given function with some restrictions, for example, f '(x)=f '(0)f(x) and f(0)=1?

4. How do i find a function that is defined everywhere but continuous nowhere?(plz explain)

Will appreciate all the help i could get, thanks.

[Justin Lazear]

1. You know that the limit of the quotient of two polynomials does not exist at a point if the denominator is zero at the point and the numerator is nonzero. So construct a polynomial denominator that has a root at x = c, i.e. (x - c)(x - r1)(x-r2)...(x - rn).

An easy way to solve for the second part is to create a polynomial that goes to 1 as x goes to c, and make this the denominator. Then all you need to do is create a polynomial that goes to L as x goes to c, and use this for the numerator.

2. Maybe I don't quite understand what you're asking with your second question, but are you asking how to find the inverse of f(x) = (a + bx)/(c + dx)? If so, you'd just use the same way as you ever would.

3. This is in general not true. It works, as far as I know, only for the exponential function and the zero function.

In order to prove it, you need more information. For instance, you need a functional equation, i.e. f(x+y) = f(x)f(y). Then you'd simply use the definition of limit, i.e.

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

4. One definition of continuity at a point is

lim_{x \to x_0}f(x) = f(x_0)

Can you create a function that never has this quality?

--J