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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.