Periodicity and Extreme Value – Problems

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