Finding polynomials with specific limits and inverse functions

[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 explanation 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?(please 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

 

[wais]

Hello! I understand that you are seeking assistance with differential calculus for an upcoming exam. I would be happy to provide some guidance and resources to help you prepare.

1. To find a nonconstant polynomial function for a given limit, you can use the fact that a polynomial function has the form f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0, where n is a nonnegative integer and a_n, a_{n-1}, ..., a_0 are constants. You can start by assuming that f(x) = a_nx^n and then solve for the other coefficients using the given limit. For example, if lim f(x)/g(x) = 13 as x approaches 2, you can set f(x) = a_nx^n and solve for a_n by setting the limit equal to 13. Then, you can continue solving for the other coefficients using the given information. Remember to also consider any restrictions on the domain of the function.

2. To find the formula for the inverse of a given function, you can use the following steps:
a) Rewrite the function in terms of y instead of x.
b) Solve for x in terms of y.
c) Switch the roles of x and y to obtain the inverse function.

For example, if f(x) = (ax + b)/(c + dx), you can rewrite it as y = (ax + b)/(c + dx) and then solve for x in terms of y. Once you have x = ..., you can switch x and y to obtain the inverse function, which would be f^{-1}(x) = ....

3. To show that an unknown function satisfies a given function with certain restrictions, you can use the definition of a derivative. In this case, you can start by taking the derivative of the given function, f(x), and then substituting in the restrictions to see if they hold. For example, for f '(x) = f '(0)f(x) with f(0) = 1, you can take the derivative of f(x) and then plug in x = 0 to see if f '(0) = 1 and if f(0) = 1.

4. A function that is defined everywhere but continuous nowhere is called a "discontinuous function." One example of such a function is the Dirichlet function, which is defined as f(x) =