Textbook Question
14–25. {Use of Tech} Areas of regions Determine the area of the given region.
The region in the first quadrant bounded by y = x/6 and y = 1−|x/2−1|
38
views
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.R.11a
{Use of Tech} Decreasing velocity A projectile is fired upward, and its velocity in m/s is given by v(t) = 200e^−t/10, for t≥0.
a. Graph the velocity function, for t≥0.
Verified step by step guidance1
Understand the velocity function: The given function is v(t) = 200e^(-t/10), where v(t) represents the velocity of the projectile at time t. The function is an exponential decay function, indicating that the velocity decreases over time.
Identify the key features of the function: The initial velocity at t = 0 is v(0) = 200e^(0) = 200 m/s. As t increases, the term e^(-t/10) decreases, causing the velocity to decrease.
Determine the behavior of the function as t approaches infinity: As t becomes very large, e^(-t/10) approaches 0, which means the velocity v(t) approaches 0 m/s.
Choose an appropriate range for t: Since the problem specifies t ≥ 0, consider a range of t values from 0 to a reasonable upper limit, such as 50 seconds, to observe the behavior of the velocity over time.
Plot the graph: On a graph, plot v(t) on the y-axis and t on the x-axis. Start at the point (0, 200) and show the curve decreasing towards the x-axis as t increases, reflecting the exponential decay of the velocity.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions in the form of f(t) = a * e^(kt), where 'e' is the base of natural logarithms. In the context of the given velocity function v(t) = 200e^(-t/10), the negative exponent indicates that the function decreases over time, which is typical for projectile motion as it loses velocity due to gravity.
Recommended video:
6:13
Exponential Functions
Graphing Functions
Graphing functions involves plotting points on a coordinate system to visualize the relationship between variables. For the velocity function v(t) = 200e^(-t/10), the graph will show how the velocity decreases as time increases, illustrating the projectile's deceleration. Understanding how to interpret and create these graphs is essential for analyzing motion.
Recommended video:
5:53Graph of Sine and Cosine Function
Velocity and Time Relationship
The relationship between velocity and time in projectile motion describes how the speed of an object changes over time. In this case, the velocity function indicates that as time progresses, the velocity of the projectile decreases exponentially, reflecting the effects of gravitational pull. This concept is crucial for understanding the dynamics of motion in physics.
Recommended video:
06:29Derivatives Applied To Velocity
Related Practice
Textbook Question
27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.
Find the area of each of the regions R₁,R₂, and R₃.
58
views
Textbook Question
Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
a. Write a single integral that gives the area of R.
152
views
Textbook Question
Comparing volumes Let R be the region bounded by y=1/x^p and the x-axis on the interval [1, a], where p>0 and a>1 (see figure). Let Vₓ and Vᵧ be the volumes of the solids generated when R is revolved about the x- and y-axes, respectively.
c. Find a general expression for Vₓ in terms of a and p. Note that p=1/2 is a special case. What is Vₓ when p=1/2?
60
views
Textbook Question
27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.
Use the shell method to find an integral, or sum of integrals, that equals the volume of the solid obtained by revolving region R₃ about the line x=3. Do not evaluate the integral.
48
views
Textbook Question
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the graph of y = 4−x² and the x-axis on the interval [−2,2] is revolved about the line x = −2. What is the volume of the solid that is generated?
80
views