For the function below: (i) What is the domain of the function? (ii) Find the derivatives of the function. (iii) What is the domain of the derivative?

Question

a) f(x) = arccos(x)

b) g(t) = sin^-1(√t)

Answer 1

f(x) = arccos(x) = inverse_cosine(x) or inv_cos(x)

(a) the domain is [-1,1] because the range of the cosine function is always between -1 <= y <= 1

(b) the derivative is -1/(sqrt(1-x^2)

(c) the argument of the square root must be positive at least. However, since it is in the denominator,

in cannot be zero in this case. so 1 - x^2 > 0

1 > x^2

x^2 < 1

x = (-1,1)

f(x) = inv_sin(sqrt(t)

(a) -1 < sqrt(t) < 1

so 0 < t < 1

(b) 1/ sqrt( 1 - t)

(c) 1-t>0 ---> 1>t