MATH 122 WORKSHEET 3 February 5, 2025 . Solve the following problems on a separate sheet. Justify your answers to earn full credit. 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'(0) = lim ( f(Ax) - f(0) Ax Ax→0 (b) Find the equation of the tangent line 1 to the graph of ƒ at P. What are the x and y intercepts of 1 ? (c) Find the equation of the line, n, through P that is perpendicular to the tangent line l. (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 the edge of 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) = -5 t² + 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, 177, 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 = lim At-0 At→0 lim ( (" h(1 + At)-h(1) At ) Evaluate this limit quotient to find the speed of the rock at time t = Express your answer in units of both m/s and ft / s. (1 meter 1 second. 3.28 feet).
MATH 122 WORKSHEET 3 February 5, 2025 . Solve the following problems on a separate sheet. Justify your answers to earn full credit. 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'(0) = lim ( f(Ax) - f(0) Ax Ax→0 (b) Find the equation of the tangent line 1 to the graph of ƒ at P. What are the x and y intercepts of 1 ? (c) Find the equation of the line, n, through P that is perpendicular to the tangent line l. (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 the edge of 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) = -5 t² + 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, 177, 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 = lim At-0 At→0 lim ( (" h(1 + At)-h(1) At ) Evaluate this limit quotient to find the speed of the rock at time t = Express your answer in units of both m/s and ft / s. (1 meter 1 second. 3.28 feet).
Trigonometry (MindTap Course List)
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Chapter6: Topics In Analytic Geometry
Section6.2: Introduction To Conics: parabolas
Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.
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Transcribed Image Text:MATH 122
WORKSHEET 3
February 5, 2025
. Solve the following problems on a separate sheet. Justify your answers to earn full credit.
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'(0) = lim (
f(Ax) - f(0)
Ax
Ax→0
(b) Find the equation of the tangent line 1 to the graph of ƒ at P.
What are the x and y intercepts of 1 ?
(c) Find the equation of the line, n, through P that is perpendicular to the tangent line l.
(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 the edge of 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) = -5 t² + 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, 177, 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
= lim
At-0
At→0
lim ( ("
h(1 + At)-h(1)
At
)
Evaluate this limit quotient to find the speed of the rock at time t =
Express your answer in units of both m/s and ft / s. (1 meter
1 second.
3.28 feet).
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