MEDIUM
Mathematics
IMPORTANT
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The function x=x-Ex is a fractional part of a number x. Prove that it is a periodic function with period 1.

Important Questions on Introduction to Mathematical Analysis

EASY
Mathematics
IMPORTANT
PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Additional Problems>Q 1.17.20.
Sketch the graph of a periodic function with period T=1 defined on the half-open interval 0,1 by the formula y=x2.
EASY
Mathematics
IMPORTANT
PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Additional Problems>Q 1.17.21.
Let us have two periodic functions fx and φx defined on a common set. Prove that if the periods of these functions are commensurate, then their sum and product are also periodic function.
EASY
Mathematics
IMPORTANT
PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Additional Problems>Q 1.17.22.
Prove that the Drichlet function λx is a periodic one but has no period.
MEDIUM
Mathematics
IMPORTANT
PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Additional Problems>Q 1.17.23.
Prove that if the function fx=sinx+cospx is periodic, then p is a rational number.
EASY
Mathematics
IMPORTANT
PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Additional Problems>Q 1.17.24.
Test the following functions for monotony : fx=x
EASY
Mathematics
IMPORTANT
PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Additional Problems>Q 1.17.24.
Test the following functions for monotony : fx=x-x
MEDIUM
Mathematics
IMPORTANT
PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Additional Problems>Q 1.17.25.
Prove that the sum of two functions increasing on a certain open interval is a function monotonically increasing on this interval. Will the difference of increasing functions be a monotonic function ?
EASY
Mathematics
IMPORTANT
PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Additional Problems>Q 1.17.26.
Give an example of a non-monotonic function that has an inverse.