CALCULUS,VOLUME 1 (OER)
16th Edition
ISBN: 2810019900790
Author: OpenStax
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Textbook Question
Chapter 2.4, Problem 174E
Prove the following functions are continuous everywhere.
174.
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A ladder 27 feet long leans against a wall and the foot of the ladder is sliding away at a constant rate of 3 feet/sec. Meanwhile, a firefighter is climbing up the ladder at a rate of 2 feet/sec. When the firefighter has climbed up 6 feet of the ladder, the ladder makes an angle of л/3 with the ground. Answer the two related
rates questions below. (Hint: Use two carefully labeled similar right triangles.)
(a) If h is the height of the firefighter above the ground, at the instant the angle of the ladder with the ground is л/3, find dh/dt=
feet/sec.
(b) If w is the horizontal distance from the firefighter to the wall, at the instant the angle of the ladder with the ground is л/3, find dw/dt=
feet/sec.
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate (in mi/h) is the distance between the cars increasing four hours later?
Step 1
Using the diagram of a right triangle given below, the relation between x, y, and z is
z²
= x²+
+12
x
Step 2
We must find dz/dt. Differentiating both sides and simplifying gives us the following.
2z
dz
dt
dx
2x.
+2y
dt
dx
dy
dz
x
+y
dt
dt
dt
2z
dy
dt
×
dx
(x+y
dt
dy
dt
An elastic rope is attached to the ground at the positions shown in the picture. The rope is being pulled up along the dotted line. Assume the units are meters.
9
ground level
Assume that x is increasing at a rate of 3 meters/sec.
(a) Write as a function of x: 0=
(b) When x=10, the angle is changing at a rate of
rad/sec.
(c) Let L be the the left hand piece of rope and R the right hand piece of rope. When x=10, is the rate of change of L larger than the rate of change of R?
○ Yes
○ No
Chapter 2 Solutions
CALCULUS,VOLUME 1 (OER)
Ch. 2.1 - For the following exercises, points P(l, 2) and...Ch. 2.1 - For the following exercises, points P(l, 2) and...Ch. 2.1 - For the following exercises, points P(l, 2) and...Ch. 2.1 - For the following exercises, points P(l, 1) and...Ch. 2.1 - For the following exercises, points P(l, 1) and...Ch. 2.1 - For the following exercises, points P(l, 1) and...Ch. 2.1 - For the following exercises, points P(4, 2) and...Ch. 2.1 - For the following exercises, points P(4, 2) and...Ch. 2.1 - For the following exercises, points P(4, 2) and...Ch. 2.1 - For the following exercises, points P(l.5, 0) and...
Ch. 2.1 - For the following exercises, points P( 1.5, 0) and...Ch. 2.1 - For the following exercises, points P( 1.5, 0) and...Ch. 2.1 - For the following exercises, points P(-1, -1) and...Ch. 2.1 - For the following exercises, points P(-1,-1) and...Ch. 2.1 - For the following exercises, points P(-1, - 1) and...Ch. 2.1 - For the following exercises, the position function...Ch. 2.1 - For the following exercises, the position function...Ch. 2.1 - For the following exercises, consider a stone...Ch. 2.1 - For the following exercises, consider a stone...Ch. 2.1 - For the following exercises, consider a rocket...Ch. 2.1 - For the following exercises, consider a rocket...Ch. 2.1 - For the following exercises, consider an athlete...Ch. 2.1 - For the following exercises, consider an athlete...Ch. 2.1 - For the following exercises, consider the...Ch. 2.1 - For the following exercises, consider the function...Ch. 2.1 - For the following exercises, consider the function...Ch. 2.1 - For the following exercises, consider the...Ch. 2.1 - For the following exercises, consider the function...Ch. 2.1 - For the following exercises, consider the function...Ch. 2.2 - For the following exercises, consider the function...Ch. 2.2 - For the following exercises, consider the function...Ch. 2.2 - For the following exercises, consider the function...Ch. 2.2 - For the following exercises, consider the function...Ch. 2.2 - For the following exercises, consider the function...Ch. 2.2 - For the following exercises, consider the function...Ch. 2.2 - For the following exercises, consider the function...Ch. 2.2 - For the following exercises, consider the function...Ch. 2.2 - [T] In the following exercises, set up a table of...Ch. 2.2 - [T] In the following exercises, set up a table of...Ch. 2.2 - [T] In the following exercises, set up a table of...Ch. 2.2 - [T] In the following exercises, set up a table of...Ch. 2.2 - [T] In the following exercises, set up a table of...Ch. 2.2 - [T] In the following exercises, set up a table of...Ch. 2.2 - [T] In the following exercises, set up a table of...Ch. 2.2 - [T] In the following exercises, set up a table of...Ch. 2.2 - In the following exercises, consider the graph of...Ch. 2.2 - In the following exercises, consider the graph of...Ch. 2.2 - In the following exercises, consider the graph of...Ch. 2.2 - In the following exercises, consider the graph of...Ch. 2.2 - In the following exercises, use the following...Ch. 2.2 - In the following exercises, use the following...Ch. 2.2 - In the following exercises, use the following...Ch. 2.2 - In the following exercises, use the following...Ch. 2.2 - In the following exercises, use the following...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, use the graph of the...Ch. 2.2 - In the following exercises, sketch the graph of a...Ch. 2.2 - In the following exercises, sketch the graph of a...Ch. 2.2 - In the following exercises, sketch the graph of a...Ch. 2.2 - In the following exercises, sketch the graph of a...Ch. 2.2 - In the following exercises, sketch the graph of a...Ch. 2.2 - Shock waves arise in many physical applications,...Ch. 2.2 - A track coach uses a camera with a fast shutter to...Ch. 2.3 - Some of the geometric formulas we take for granted...Ch. 2.3 - Some of the geometric formulas we take for granted...Ch. 2.3 - Some of the geometric formulas we take for granted...Ch. 2.3 - Some of the geometric formulas we take for granted...Ch. 2.3 - Some of the geometric formulas we take for granted...Ch. 2.3 - In the following exercises, use the limit Laws to...Ch. 2.3 - In the following exercises, use the limit laws to...Ch. 2.3 - In the following exercises, use the limit laws to...Ch. 2.3 - In the following exercises, use the limit laws to...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - ]In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, use direct...Ch. 2.3 - In the following exercises, assume that...Ch. 2.3 - In the following exercises, assume that...Ch. 2.3 - In the following exercises, assume that...Ch. 2.3 - In the following exercises, assume that...Ch. 2.3 - In the following exercises, assume that...Ch. 2.3 - In the following exercises, assume that...Ch. 2.3 - In the following exercises, assume that...Ch. 2.3 - In the following exercises, assume that...Ch. 2.3 - [T] In the following exercises, use a calculator...Ch. 2.3 - [T] In the following exercises, use a calculator...Ch. 2.3 - [T] In the following exercises, use a calculator...Ch. 2.3 - In the following exercises, use the following...Ch. 2.3 - In the following exercises, use the following...Ch. 2.3 - In the following exercises, use the following...Ch. 2.3 - yIn the following exercises, use the following...Ch. 2.3 - In the following exercises, use the following...Ch. 2.3 - In the following exercises, use the following...Ch. 2.3 - In the following exercises, use the following...Ch. 2.3 - In the following exercises, use the following...Ch. 2.3 - For the following problems, evaluate the limit...Ch. 2.3 - For the following problems, evaluate the limit...Ch. 2.3 - For the following problems, evaluate the limit...Ch. 2.3 - [T] In physics, the magnitude of an electric field...Ch. 2.3 - [T] The density of an object is given by its mass...Ch. 2.4 - For the following exercises, determine the...Ch. 2.4 - For the following exercises, determine the...Ch. 2.4 - For the following exercises, determine the...Ch. 2.4 - For the following exercises, determine the...Ch. 2.4 - For the following exercises, determine the...Ch. 2.4 - For the following exercises, determine the...Ch. 2.4 - For the following exercises, determine the...Ch. 2.4 - For the following exercises, determine the...Ch. 2.4 - For the following exercises, decide if the...Ch. 2.4 - For the following exercises, decide if the...Ch. 2.4 - For the following exercises, decide if the...Ch. 2.4 - For the following exercises, decide if the...Ch. 2.4 - For the following exercises, decide if the...Ch. 2.4 - For the following exercises, decide if the...Ch. 2.4 - In the following exercises, find the value(s) of k...Ch. 2.4 - In the following exercises, find the value(s) of k...Ch. 2.4 - In the following exercises, find the value(s) of k...Ch. 2.4 - In the following exercises, find the value(s) of k...Ch. 2.4 - In the following exercises, find the value(s) of k...Ch. 2.4 - In the following exercises, use the Intermediate...Ch. 2.4 - In the following exercises, use the Intermediate...Ch. 2.4 - In the following exercises, use the Intermediate...Ch. 2.4 - In the following exercises, use the Intermediate...Ch. 2.4 - Consider the graph of the function y=f(x) shown in...Ch. 2.4 - Let f(x)={3x,x1x3,x1 . Sketch the graph of f. Is...Ch. 2.4 - Let f(x)=x41x21forx1,1 . a. Sketch the graph of f....Ch. 2.4 - Sketch the graph of the function y=f(x) with...Ch. 2.4 - Sketch the graph of the function y=f(x) with...Ch. 2.4 - In the following exercises, suppose y=f(x) is...Ch. 2.4 - In the following exercises, suppose y=f(x) is...Ch. 2.4 - Determine whether each of the given statements is...Ch. 2.4 - Determine whether each of the given statements is...Ch. 2.4 - Determine whether each of the given statements is...Ch. 2.4 - Determine whether each of the given statements is...Ch. 2.4 - Determine whether each of the given statements is...Ch. 2.4 - Determine whether each of the given statements is...Ch. 2.4 - Determine whether each of the given statements is...Ch. 2.4 - [T] The following problems consider the scalar...Ch. 2.4 - [T] The following problems consider the scalar...Ch. 2.4 - [T] The following problems consider the scalar...Ch. 2.4 - [T] After a certain distance D has passed, the...Ch. 2.4 - As the rocket travels away from Earth’s surface,...Ch. 2.4 - wqProve the following functions are continuous...Ch. 2.4 - Prove the following functions are continuous...Ch. 2.4 - Prove the following functions are continuous...Ch. 2.5 - In the following exercises, write the appropriate ...Ch. 2.5 - In the following exercises, write the appropriate ...Ch. 2.5 - In the following exercises, write the appropriate ...Ch. 2.5 - In the following exercises, write the appropriate ...Ch. 2.5 - The following graph of the function f satisfies...Ch. 2.5 - The following graph of the function f satisfies...Ch. 2.5 - The following graph of the function f satisfies...Ch. 2.5 - The following graph of the function f satisfies...Ch. 2.5 - The following graph of the function f satisfies...Ch. 2.5 - The following graph of the function f satisfies...Ch. 2.5 - [T] In the following exercises, use a graphing...Ch. 2.5 - [T] In the following exercises, use a graphing...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - In the following exercises, use the precise...Ch. 2.5 - An engineer is using a machine to cut a flat...Ch. 2.5 - Use the precise definition of limit to prove that...Ch. 2.5 - Using precise definitions of limits, prove that...Ch. 2.5 - Using precise definitions of limits, prove that...Ch. 2.5 - Using precise definitions of limits, prove that...Ch. 2.5 - Using the function from the previous exercise, use...Ch. 2.5 - limxa(f(x)g(x))=LMCh. 2.5 - limxa[cf(x)]=cL for any real constant c (Hint....Ch. 2.5 - ...Ch. 2 - wTrue or False. In the following exercises,...Ch. 2 - True or False. In the following exercises, justify...Ch. 2 - True or False. In the following exercises, justify...Ch. 2 - True or False. In the following exercises, justify...Ch. 2 - Using the graph, find each limit or explain why...Ch. 2 - In the following exercises, evaluate the limit...Ch. 2 - In the following exercises, evaluate the limit...Ch. 2 - In the following exercises, evaluate the limit...Ch. 2 - In the following exercises, evaluate the limit...Ch. 2 - wIn the following exercises, evaluate the limit...Ch. 2 - In the following exercises, evaluate the limit...Ch. 2 - In the following exercises, evaluate the limit...Ch. 2 - In the following exercises, evaluate the limit...Ch. 2 - In the following exercises, evaluate the limit...Ch. 2 - In the following exercises, evaluate the limit...Ch. 2 - In the following exercises, use the squeeze...Ch. 2 - In the following exercises, use the squeeze...Ch. 2 - In the following exercises, use the squeeze...Ch. 2 - In the following exercises, determine the value of...Ch. 2 - In the following exercises, determine the value of...Ch. 2 - In the following exercises, use the precise...Ch. 2 - In the following exercises, use the precise...Ch. 2 - A ball is thrown into the air and the vertical...Ch. 2 - A particle moving along a line has a displacement...Ch. 2 - From the previous exercises, estimate the...
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