Periodicity and Extreme Value – Problems
Find the periods for the given functions.
1. Cos (3x + 5) + 7
Solution: Given that
Cos (3x + 5) + 7
Period of cosx = 2π
Period = \(\frac{period\ of\ \cos x}{[coefficient\ of\ x]}=\frac{2\pi }{3}\)
2. Tan5x
Solution: Given that
Tan5x
f(x) = tan5x
Period of tanx = π
Period = \(\frac{period\ of\ \cos x}{[coefficient\ of\ x]}=\frac{\pi }{5}\)
3. |sinx|
Solution: Given that,
|sinx|
Period = π
|sin(π + x)| = |(-sinx)| = sinx
4. tan (x + 4x + 9x + …… + n²x) (n any positive integer)
Solution: Given that
tan (x + 4x + 9x + …… + n²x)
1 + 2² + 3² + …… + n² = \(\frac{n(n+1)(2n+1)}{6}\)
tan(1 + 4 + 9 + …… + n²)x
tan[n(n+1)(2n+1)/6]x
period = \(\frac{\pi }{\frac{n(n+1)(2n+1)}{6}}=\frac{6\pi }{n(n+1)(2n+1)}\)
5. sketch the region enclosed by y = sinx, y = cosx and x – axis in the interval [0, π]
Solution: y = sinx
|
X |
0 | π/4 | π/2 | 3π/4 | π |
| Y | 0 | 1/√2 | 1 | 1/√2 |
0 |
y = cosx
|
X |
0 | π/4 | π/2 | 3π/4 | π |
| Y | 1 | 1/√2 | 0 | -1/√2 |
-1 |
