Textbook Question
By computing the first few derivatives and looking for a pattern, find the following derivatives.
a. d⁹⁹⁹/dx⁹⁹⁹ (cos x)
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.4.15a
The accompanying figure shows the velocity v = ds/dt = f(t) (m/sec) of a body moving along a coordinate line.
a. When does the body reverse direction?
Verified step by step guidance1
To determine when the body reverses direction, we need to find when the velocity changes sign. This occurs when the velocity function v(t) crosses the horizontal axis (v = 0).
Examine the graph of v(t) to identify the points where the curve crosses the t-axis. These points are where the velocity is zero.
From the graph, observe that the velocity crosses the t-axis at approximately t = 2 seconds, t = 6 seconds, and t = 9 seconds.
At these points, the velocity changes from positive to negative or from negative to positive, indicating a reversal in direction.
Thus, the body reverses direction at t = 2 seconds, t = 6 seconds, and t = 9 seconds.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity and Direction
Velocity is a vector quantity that indicates the rate of change of position with respect to time. When analyzing motion, the sign of the velocity determines the direction of movement. A positive velocity indicates motion in one direction, while a negative velocity indicates motion in the opposite direction. Understanding how velocity changes over time is crucial for determining when an object reverses its direction.
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Derivatives Applied To Velocity
Critical Points and Zeroes
Critical points occur where the velocity function is zero or undefined. These points are essential for analyzing motion because they indicate potential changes in direction. In the context of the graph, when the velocity crosses the time axis (where v = 0), it signifies that the body may be reversing its direction. Identifying these zeroes helps in understanding the overall motion of the object.
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04:50Critical Points
Graph Interpretation
Interpreting graphs is a fundamental skill in calculus, particularly when analyzing functions like velocity. The graph of velocity versus time provides visual insights into how the velocity changes over time. By examining the graph, one can identify intervals of positive and negative velocity, as well as points where the velocity is zero, which are critical for determining when the body reverses direction.
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06:15Graphing The Derivative
Related Practice
Textbook Question
Faster than a calculator Use the approximation (1 + x)ᵏ ≈ 1 + kx to estimate the following.
a. (1.0002)⁵⁰
Textbook Question
Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.
a. How fast is the boat approaching the dock when 10 ft of rope are out?
Textbook Question
Consider the function f graphed here. The domain of f is the interval [−4, 6] and its graph is made of line segments joined end to end.
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b. Graph the derivative of f. The graph should show a step function.
Textbook Question
Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is
A = (1/2) ab sinθ.
a. How is dA/dt related to dθ/dt if a and b are constant?
Textbook Question
a. Let f(x) be a function satisfying |f(x)| ≤ x² for −1 ≤ x ≤ 1. Show that f is differentiable at x = 0 and find f′(0).