2024/25 Semester 1
MH1810 Math 1
Take Home Test
Version L
Answer All Questions
1. Use the Mean Value Theorem to show the following estimate of the sixth root of 65:
2. Find the following limits
3. Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 9 cm. Give your answer in exact form.
4. Express the following as a definite integral f (x) dx and find its exact value. (Express your answer in the form. a ln b).
5. (a) Show that where A; B; C are to be determined.
(b) Use integration by parts to establish the following reduction formula
Deduce that
6. Let f be a function defined on [0; π] ; where
Show that xf (x) = sin x for all x 2 [0; π] : Hence find the volume when the region bounded by y = f (x) and x-axis is revolved about the y-axis by 2π radians.