Domanda del libro di testo
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = √(3x + 3); P(2,3)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 18b
Position, velocity, and acceleration Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left?
f(t) = 18t-3t²; 0 ≤ t ≤ 8
Guida verificata passo dopo passo1
Step 1: Understand the relationship between position, velocity, and acceleration. The position function s = f(t) describes the location of an object at time t. The velocity function v(t) is the derivative of the position function, representing the rate of change of position with respect to time.
Step 2: Differentiate the position function f(t) = 18t - 3t^2 to find the velocity function v(t). Use the power rule for differentiation: if f(t) = at^n, then f'(t) = n*at^(n-1).
Step 3: Apply the power rule to each term in f(t). The derivative of 18t is 18, and the derivative of -3t^2 is -6t. Therefore, the velocity function is v(t) = 18 - 6t.
Step 4: Determine when the object is stationary by setting the velocity function equal to zero and solving for t. This will give the time(s) when the object is not moving.
Step 5: Analyze the sign of the velocity function to determine when the object is moving to the right (v(t) > 0) and moving to the left (v(t) < 0). This involves solving inequalities based on the velocity function.
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Position Function
The position function, denoted as s = f(t), describes the location of an object at any given time t. In this case, the function f(t) = 18t - 3t² represents the position of the object in feet. Understanding this function is crucial as it provides the basis for determining both velocity and acceleration, which are derived from the position function.
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Percorso guidato
5:20
Relations and Functions
Velocity Function
The velocity function is the first derivative of the position function with respect to time, represented as v(t) = f'(t). It indicates the rate of change of position, showing how fast and in which direction the object is moving. For the given position function, calculating the derivative will yield the velocity function, which can then be analyzed to determine when the object is stationary (v(t) = 0), moving to the right (v(t) > 0), or moving to the left (v(t) < 0).
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Percorso guidato
06:29Derivatives Applied To Velocity
Graphing Functions
Graphing functions involves plotting the values of a function on a coordinate system to visualize its behavior. For the velocity function derived from the position function, the graph will help identify key features such as intercepts, maxima, and minima. Analyzing the graph allows for a clear understanding of the object's motion over time, including when it changes direction or comes to a stop.
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Percorso guidato
5:53Graph of Sine and Cosine Function
Pratica correlata
Domanda del libro di testo
5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = √x²+1
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Domanda del libro di testo
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = 4/x2; P(-1,4)
189
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Domanda del libro di testo
5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = sin x⁵
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Domanda del libro di testo
Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
On what intervals is the speed increasing?
f(t) = 18t - 3t2; 0 ≤ t ≤ 8
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Domanda del libro di testo
Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
On what intervals is the speed increasing?
f(t) = 2t2 - 9t + 12; 0 ≤ t ≤ 3
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