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Elementary Calculus: Integers, Derivatives, 14 Questions

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1. Let f:Z->Z , where Z is the set of integers and f(x)=x^5+101
Is f(x) a one-to-one function?

2. Prove that f(x)=4x-3 is one-to-one function

3. The ceiling function maps every real number to the smallest integer greater than
or equal to that number, f:x->[x], where [x] is the smallest integer greater
than or equal to x. Is this function one-to-one? List five numbers that have
the same image

4. Find lim n^3 + n when n-> ?
n^2 - 1

5. Find lim 3x^7 + x^3 +17 , as x-> ?
2x^7 + X^2 + 1

6. Find lim x^2008 + 1 , as x-> ?
x^2009 + 1

7. Find lim x^4 + 111,111 , as x-> 0
x^3 + 111,111,111

8. Prove that f(x)=x^2-4 is a continuous function

9. Find derivatives of the following functions using differentiation rules

1. f(x)= 9x+5
2. f(x)= 3e^x
3. f(x)= 3x^3 + 4
4. f(x)= x^4 + 3x^2 + 4x - 31
5. f(x)= (2x + 3)e^x
6. f(x)= (2x-1)e^(x-3)
7. f(x)= x(x-2)^2 =x(x^2 - 4x + 4) = x^3 - 4x^2 + 4x

10. Find the derivative of the following function at x0 = -1

f(x)=3x^2 + x + 1

11. Find differentials of the following functions:
1. f(x)= 3x^2 + x + 1
2. f(x)= e^(x-1)
3. f(x)= -e^x + x^3 - 4

12. Let f(x)= x and g(x)=x^2 , where xER
Find f?g and g?f as well as the domain and range of these functions.
13. Prove using the definition that lim 1/n^2 = 0 , when n-> ?
14. Find the derivative of f(x)=1/x by using the definition of the derivative

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Solution Summary

Elementary calculus for integers and derivatives are examined.