Skip to main content
Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.31b
Chapter 2, Problem 2.31b

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

Verified step by step guidance
1
insert step 1: Understand that the instantaneous velocity of the projectile is given by the derivative of the position function s(t).
insert step 2: Differentiate the position function s(t) = -16t^2 + 128t + 192 with respect to t to find the velocity function v(t).
insert step 3: Set the velocity function v(t) equal to zero to find the time when the projectile has an instantaneous velocity of zero.
insert step 4: Solve the equation from step 3 for t to find the value of t = a.
insert step 5: Verify that the value of t = a is within the given interval 0 ≤ t ≤ 9.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Position Function

The position function describes the location of an object at any given time. In this case, s(t) = -16t^2 + 128t + 192 represents the height of a projectile over time, where 't' is the time in seconds. Understanding this function is crucial for analyzing the projectile's motion and determining its behavior at different time intervals.
Recommended video:
5:20
Relations and Functions

Instantaneous Velocity

Instantaneous velocity is the rate of change of the position function with respect to time, represented mathematically as the derivative of the position function, s'(t). For the given position function, finding the time when the instantaneous velocity is zero involves calculating the derivative and solving for t when s'(t) = 0, indicating the moment the projectile reaches its peak height.
Recommended video:
06:29
Derivatives Applied To Velocity

Critical Points

Critical points occur where the derivative of a function is zero or undefined, indicating potential maxima, minima, or points of inflection. In the context of the projectile's motion, identifying the critical points of the velocity function helps determine when the projectile stops ascending and starts descending, which is essential for understanding its overall trajectory.
Recommended video:
04:50
Critical Points
Related Practice
Textbook Question

Complete the following steps for the given functions. 


b. Find the vertical asymptotes of ff (if any).


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

Textbook Question

Assume you invest \(250 at the end of each year for 10 years at an annual interest rate of rr. The amount of money in your account after 10 years is given by A(r)=250((1+r)10−1)rA\left(r\right)=\frac{250\left(\left(1+r\right)^{10}-1\right)}{r}. Assume your goal is to have \)3500 in your account after 10 years.


b. Use a calculator to estimate the interest rate required to reach your financial goal.

Textbook Question

Let g(x)={x2+xif x<1aif x=13x+5if x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)

b. Determine the value of aa for which gg is continuous from the right at 11. 

1
views
Textbook Question

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1^+ f(x)

Textbook Question

Complete the following steps for the given functions. 


b. Find the vertical asymptotes of f (if any).


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

1
views
Textbook Question

Complete the following sentences in terms of a limit.


b. A function is continuous from the right at a if _____ .