MMA 102 Calculus I Cat 2 Take Home PDF

Tom Mboya University, Faculty of Physical and Biological Sciences MMA 102: CALCULUS I CAT 2 - TAKE HOME Instruction: Attempt ALL questions. 1 Let f : R −→ R and g : R −→ R be given by f (x) = log x...

Tom Mboya University, Faculty of Physical and Biological Sciences MMA 102: CALCULUS I CAT 2 - TAKE HOME Instruction: Attempt ALL questions. 1 Let f : R −→ R and g : R −→ R be given by f (x) = log x, and g(x) = 1−x 2. (i)What are the domains of the functions f and g. (ii)Determine the composite function (f ◦ g)(x), hence or otherwise, evaluate (f ◦ g)(−1). (a) (i)State the Mean Value Theorem (ii)Use the Mean Value Theorem to prove that | sin a − sin b| ≤ |a − b| for all real values of a and b where a ̸= b. (b)Suppose that f : R → R is a differentiable function with f (0) = 5 and −1 ≤ f ′ (x) ≤ 3 for all x. Show that −5 ≤ f (10) (c)A car passes a camera at a point A on the toll road with speed 50km/h. One hour later the same car passes a camera a dy (a)If a point traces the circle x2 + y 2 = 25 and if dx dt = 4 when the point reaches (3, 4), find dt at this point. 3 2 (b)A body moves in a straight line according to the law of motion S = t − 4t − 3t. Find its acceleration at each instant w (c)By differentiating x2 − y 2 = 1 implicitly, show that y ′′ = − y13 (d)Find the points on the curve y = 2x3 − 3x2 − 12x + 20 where the tangent is parallel to the x-axis. dy Find dx in the√ following functions 2 (a)y = cos(sin x2 + 1) (b)x2 = x−y x+y (c)y = cos−1 2x (d)y = x3 xln 2x (e)y = xx 2 2 (x−3)4 (f)y = (x +9)x2 +2