COMSATS Institute of Information Technology, Islamabad MTH104 SP13 Calculus and Analytic Geometry Assignment 1: Functions, limits and derivatives The due date for this assignment is Thursday, April 4, 2013. Warning: Failure to show complete working will result in a loss of marks 1. Let H : R ! f0; 1g be a function dened as follows H(t) = 0 t 0 1 t > 0 This function is known as the switching function. In other literature, it is also known as the Heaviside Function or the Unit Step Function. As implied by the name, the purpose of this function is to “switch”. Before t = 0, the value of H is zero, indicating an OFF and after t = 0, the value of this function jumps from 0 to 1, indicating an ON. Let V (t) = t sin 2t be the voltage function of an alternating current passing through a given electric circuit after time t for t 2 R. If a switch connected to the given circuit is switched on after t = 0, construct a function inidcating the switch, using V and H: [1 marks]. 2. Let f : R ! R and g : R ! R be real-valued functions dened by f(x) = x3 + 7 and g(x) = 7x + 9: Determine [3+3+3 marks] (a) f f (b) g f (c) f g 3. Let f and g be the same as in question 2. Then, determine f??1(x) and g??1(x): State the changes in the domain and range of each function. [4 marks] 4. Solve for x without using a calculator. Show your working. [3+3+3 marks] e2x + 4ex + 4 = 0 cos 2x + sin x + 2 cos x + 1 = 0 tan4 x ?? 2 + 2 8 tan2 x + 4 256 = 0 5. Let f(x) = x2??1+cos(x2) . Then, determine [2+2+1+2+2 marks] (a) lim x!p??f(x) (b) lim x!p+f(x) 1
(c) lim x!pf(x) (d) Is f a continuous function? State the domain of f under which it is continuous (e) Can any values of f be dened so that f becomes continuous on R? If so, state such values. 6. Determine all positive integers m such that the function f(x) = 1 sin(2m+1)x is undefned [3 marks] 7. Find the equation of the tangent line of the unit circle centered at (5; 7) at x = 3=2 [5 marks] 2