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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.6.40b
Chapter 3, Problem 3.6.40b

Velocity of a car The graph shows the position s=f(t) of a car t hours after 5:00 P.M. relative to its starting point s=0,where s is measured in miles. <IMAGE>
b. At approximately what time is the car traveling the fastest? The slowest?

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1
To determine when the car is traveling the fastest or the slowest, we need to analyze the graph of the position function s=f(t). The speed of the car is given by the derivative of the position function, which is the velocity v(t)=f'(t).
Identify the points on the graph where the slope of the tangent line is the steepest. The steepest positive slope indicates the fastest speed, while the steepest negative slope indicates the slowest speed.
Look for the points on the graph where the slope changes from positive to negative or vice versa. These points are typically where the velocity is zero, indicating a change in direction or a stop.
Estimate the time t at which these slopes occur by observing the graph. The time when the slope is steepest positive corresponds to the fastest speed, and the time when the slope is steepest negative corresponds to the slowest speed.
Consider the context of the problem: the graph represents the position relative to the starting point at 5:00 P.M. Use this information to convert the time t into actual clock time for your final answer.

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Key Concepts

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

Velocity and Speed

Velocity is the rate of change of position with respect to time, represented mathematically as the derivative of the position function, s=f(t). Speed, a scalar quantity, refers to how fast an object is moving regardless of direction. Understanding how to interpret the graph of position versus time is crucial for determining when the car is traveling fastest or slowest.
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Derivatives Applied To Velocity

Critical Points

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima or minima. In the context of the car's velocity, these points on the graph of the position function can help identify when the car is at its fastest or slowest speeds. Analyzing these points allows us to determine changes in the car's motion.
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Critical Points

Graph Interpretation

Interpreting graphs involves analyzing the shape and features of the graph to extract meaningful information. For the position-time graph of the car, the slope at any point indicates the car's velocity. A steeper slope corresponds to higher speeds, while a flatter slope indicates slower speeds, which is essential for answering the question about the car's speed at different times.
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Related Practice
Textbook Question

Use the graph of g in the figure to do the following. <IMAGE>

b. Find the values of x in (-2,2) at which g is not differentiable.

Textbook Question

Use a graphing utility to plot the curve and the tangent line.

y = cos x / 1−cos x; x = π/3

Textbook Question

The energy (in joules) released by an earthquake of magnitude M is given by the equation E=25,000 ⋅ 101.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)

Compute dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)

Textbook Question

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

Textbook Question

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

limx→04+sin(x)−2x{\(\displaystyle\]\lim\)_{x\(\to\)0}}\(\frac{\sqrt{4+\sin\left(x\right)}\)-2}{x}

Textbook Question

45–50. Tangent lines Carry out the following steps. <IMAGE>

b. Determine an equation of the line tangent to the curve at the given point.

x⁴-x²y+y⁴=1; (−1, 1)

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