MAT 201 - Calculus One Wolfram | Alpha Lab Five Topic: Tangents, Rates of Change, and Derivatives Instructions: Each student should read the lab carefully and completely. Be sure to complete each step and give your response to each question. Students should use Wolfram | Alpha (www.wolframalpha.com) to assist in the computations required for the lab such as plots, evaluations, etc. Each student should submit a Word (or similar word processing) document that displays the graphs and responses for the lab assignment. Please make sure to include your name and Z number at the top of your lab document. Purpose The purpose of this lab is to calculate slope(s) and rates of change in order to find the equation of a tangent line to a curve. The lab will utilize the concept of a limit to define the value of the slope of tangent line. This lab will focus on using one of the definitions for slope given below. Finally, the lab will verify graphically that the tangent lines are correct. slope = m = lim f(x) -J(@) = f(a) OR slope =m= lim J(ath)-f(a) = f'(a) x->a x - a h - h Action 1. Using one of the slope formulas given above, find the rate of change of g(x)= 1 2 atx = 2. Be sure to show the limit expression you evaluated on your lab assignment document. 2. Using one of the formulas given above, find the derivative (slope of tangent line) of f(x)= x sin(x) at x = nt/4. Be sure to show the limit expression you evaluated on your lab assignment document 3. Given q(x)= 15-2x , find the equations of the tangent lines to q(x) at (1/2 , 1/2) and (-2, 1/3). Graph the curve and both tangents together. Be sure to show the limit expression you evaluated to find the slope on your lab assignment document. {Remember the point-slope form of a line is: y - y1 = m(x - x1)} 4. Show that the curve y = 6x' + 5x -3 has no tangent line with slope 4. 5. Find the equation of the Normal Line to the curve at the given point. (The Normal Line to a curve C is at a point P is, by definition, the line that passes through P, and is perpendicular to the tangent line to C at P.) Sketch the curve and the Normal line together on a graph. C(x)= Vx P= (-8,-2) Tips and Suggestions 1) To specify a particular domain for a plot use the following syntax as the input: Plot sin (x) for x = -4.6 to 4. 6 2) To plot two or more graphs together simply use the word "and". Input into Alpha looks like: plot (1/(2-3x)) and e^(4x)