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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.R.11a
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 guidance
1
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.

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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.
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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.
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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.
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Related Practice
Textbook Question

2–3. Displacement, distance, and position Consider an object moving along a line with the following velocities and initial positions. Assume time t is measured in seconds and velocities have units of m/s.


d. Determine the position function s(t) using the Fundamental Theorem of Calculus (Theorem 6.1). Check your answer by finding the position function using the antiderivative method.


v(t) = 12t²-30t+12, for 0 ≤ t ≤ 3; s(0)=1

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.

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Textbook Question

Force on a dam Find the total force on the face of a semicircular dam with a radius of 20 m when its reservoir is full of water. The diameter of the semicircle is the top of the dam.

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?

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Textbook Question

35-38. Area and volume Let R be the region in the first quadrant bounded by the graph of

Find the area of the region R.

Textbook Question

Fuel consumption A small plane in flight consumes fuel at a rate (in gal/min) given by

R'(t) ={ 4t^{1/3} if 0 ≤ t ≤ 8 (take-off)

2 if t> 0 (cruising)

a. Find a function R that gives the total fuel consumed, for 0≤t≤8.