1. Let f(x)=x - 2x +1. (a) Find the slope of the graph of y =...

 

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1. Let f(x)=x² - 2x +1. (a) Find the slope of the graph of y = f(x) at the point P = (0.1) by directly evaluating the limit: f(Ax)-f(0)\ Ar f'(0) = lim (b) Find the equation of the tangent line I to the graph off at P. What are the x and y intercepts of !? (c) Find the equation of the line, n, through P that is perpendicular to the tangent line 1. (Line n is called the normal line to the graph of f at P.) (d) Sketch a careful graph that displays: the graph of y = f (x), its vertex point, its tangent and normal lines at point P, and the x and y intercepts of these lines. Bonus: Find the coordinates of the second point, Q, (QP), at which the normal line n intersects the graph of f. 2. A rock is thrown vertically upward with an initial velocity of 20 m/s from a bridge that is 25 meters above a river bed. Based on Newton's Laws of motion, the height of the rock (above the river) t seconds after being launched is given by: h(t) = -5t² + 20 t + 25 meters. (a) Determine the exact time, t, at which the rock hits the river by solving the equation h (t) = 0 for t. (b) Sketch a graph of y=h(t) for 0≤t≤t, seconds. (c) Using your graph from (b), deduce the maximum height attained by the rock above the river. (d) Determine the average velocity of the rock, from time t=1 to t= 2 seconds. Does this value overestimate or underestimate the instantaneous velocity of the rock at time t = 1 second? Explain your reasoning. (e) The instantaneous velocity of the rock at time t = 1 second is given by h'(1), where h'(1) lim ( 41-0 lim at-0 (h(1+At)-h(1)) ()-h(¹)). Evaluate this limit quotient to find the speed of the rock at time t = 1 second. Express your answer in units of both m/s and ft/s. (1 meter = 3.28 feet) 1. Let f(x)=x² - 2x +1. (a) Find the slope of the graph of y = f(x) at the point P = (0.1) by directly evaluating the limit: f(Ax)-f(0)\ Ar f'(0) = lim (b) Find the equation of the tangent line I to the graph off at P. What are the x and y intercepts of !? (c) Find the equation of the line, n, through P that is perpendicular to the tangent line 1. (Line n is called the normal line to the graph of f at P.) (d) Sketch a careful graph that displays: the graph of y = f (x), its vertex point, its tangent and normal lines at point P, and the x and y intercepts of these lines. Bonus: Find the coordinates of the second point, Q, (QP), at which the normal line n intersects the graph of f. 2. A rock is thrown vertically upward with an initial velocity of 20 m/s from a bridge that is 25 meters above a river bed. Based on Newton's Laws of motion, the height of the rock (above the river) t seconds after being launched is given by: h(t) = -5t² + 20 t + 25 meters. (a) Determine the exact time, t, at which the rock hits the river by solving the equation h (t) = 0 for t. (b) Sketch a graph of y=h(t) for 0≤t≤t, seconds. (c) Using your graph from (b), deduce the maximum height attained by the rock above the river. (d) Determine the average velocity of the rock, from time t=1 to t= 2 seconds. Does this value overestimate or underestimate the instantaneous velocity of the rock at time t = 1 second? Explain your reasoning. (e) The instantaneous velocity of the rock at time t = 1 second is given by h'(1), where h'(1) lim ( 41-0 lim at-0 (h(1+At)-h(1)) ()-h(¹)). Evaluate this limit quotient to find the speed of the rock at time t = 1 second. Express your answer in units of both m/s and ft/s. (1 meter = 3.28 feet)