# MH1810 Math Take Home Test Home Work, NTU, Singapore: Use Intermediate Value Theorem And Mean Value Theorem To Show That The Graphs Of Y = Cos X And Y = X + 2 Intersect Exactly Once

Questions 1,2,3 carry 2 marks each and question 4 carries 3

marks. Total 9 marks. Answer ALL questions.

**1.** Use the intermediate value theorem and mean value theorem to show that

the graphs of y = cos x and y = x + 2 intersect exactly once.

**2.** A 2.6 m ladder is leaning against a wall when its base starts to slide

away. By the time the base is 2.4 m from the wall, the base is moving

away at the rate of 1 ms1 : At what rate is the angle between the ladder and the ground changing at this moment?

3. Express the following as a deÖnite integral R 1 0 f (x) dx and Önd its exact value. (Express your answer in terms of )