Chemistry 1S - Prof Paul May Calculus Question for January Exam 2011 Answer parts (a) to (e) all Describe and sketch the main features of the function = cos(ax) where u(x) is a constant. An accurate plot on graph-paper is required. a not (5 marks) Describe and sketch the main features of the function = exp(–bx) v(x) where is a constant. An accurate plot on graph-paper is required. b not (5 marks) The function below, which is a product of the two functions in (a) and (b), is often used to represent damped simple-harmonic-motion: = cos(ax) exp( y(x) –bx) (c) Differentiate with respect to y(x) x. (6 marks) (d) Find a general expression involving and for the stationary points of a b y(x). (6 (e) Sketch in your answerbooks for positive values of labelling the y(x) x, important parts of the curve. An accurate plot on graph-paper is not required. (8 marks) Answers = cos(ax) is a cosine function, a function which repeats itself with a period u i.e. of 2/a. It cycles between a maximum amplitude of +1 at and = 2/a, and a x=0 x minimum amplitude of 1 at The function = 0 at /2a and 3/2a. x=/a. u = exp( is an exponential decay function, with decay constant = It has a v bx) b. maximum value of 1 at and then drops away exponentially with increasing x=0, x at a rate determined by the magnitude of The function only reaches 0 at b. x=∞. Product Rule: = cos(ax)( + (exp( [4 marks] bexp(-bx)) bx)( asin(ax))
Chemistry 1S - Prof Paul May Calculus Question for January Exam 2011 Answer all parts (a) to (e) Describe and sketch the main features of the function u(x) = cos(ax) where a is a constant. An accurate plot on graph-paper is not required. (5 marks) Describe and sketch the main features of the function v(x) = exp(–bx) where b is a constant. An accurate plot on graph-paper is not required. (5 marks) The function below, which is a product of the two functions in (a) and (b), is often used to represent damped simple-harmonic-motion: y(x) = cos(ax) exp( –bx) (c) Differentiate y(x) with respect to x. (6 marks) (d) Find a general expression involving a and b for the stationary points of y(x). (6 (e) Sketch y(x) in your answerbooks for positive values of x, labelling the important parts of the curve. An accurate plot on graph-paper is not required. (8 marks) Answers u = cos(ax) is a cosine function, i.e. a function which repeats itself with a period of 2/a. It cycles between a maximum amplitude of +1 at x=0 and x = 2/a, and a minimum amplitude of 1 at x=/a. The function u = 0 at /2a and 3/2a. v = exp( bx) is an exponential decay function, with decay constant = b. It has a maximum value of 1 at x=0, and then drops away exponentially with increasing x at a rate determined by the magnitude of b. The function only reaches 0 at x=∞. Product Rule: = cos(ax)( bexp(-bx)) + (exp( bx)( asin(ax)) [4 marks]