Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x³/3 + x²/2 + x/4
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 3.1.30
Rates of Change
Speed of a rocket At t sec after liftoff, the height of a rocket is 3t² ft. How fast is the rocket climbing 10 sec after liftoff?
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To find how fast the rocket is climbing, we need to determine the rate of change of the height with respect to time, which is the derivative of the height function.
The height of the rocket as a function of time is given by h(t) = 3t². We need to find the derivative of this function, h'(t), to get the velocity function.
Differentiate h(t) = 3t² with respect to t. Using the power rule, the derivative of t² is 2t, so h'(t) = 3 * 2t = 6t.
Now that we have the velocity function h'(t) = 6t, we can find the velocity at a specific time by substituting t = 10 seconds into this derivative.
Substitute t = 10 into h'(t) = 6t to find the velocity of the rocket at 10 seconds. This will give you the rate at which the rocket is climbing at that moment.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures the rate at which the function's value changes as its input changes. In the context of the rocket's height, the derivative of the height function with respect to time will give us the rocket's speed at any given moment.
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Derivatives
Function of Time
In this problem, the height of the rocket is expressed as a function of time, specifically h(t) = 3t². Understanding how to interpret and manipulate this function is crucial for determining the rocket's height and speed at specific time intervals.
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6:13Exponential Functions
Application of the Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. In this scenario, while the height function is straightforward, recognizing how to apply differentiation rules effectively is essential for calculating the speed of the rocket at the specified time.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.
y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2
Textbook Question
In Exercises 53 and 54, find dr/ds.
r cos 2s + sin²s = π
Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
w = 3z⁷ − 7z³ + 21z²
Textbook Question
Approximation Error
In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find
a. the change Δf = f(x₀ + dx) − f(x₀);
b. the value of the estimate df = fʹ(x₀) dx; and
c. the approximation error |Δf − df|.
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f(x) = x² + 2x, x₀ = 1, dx = 0.1
Textbook Question
A melting ice layer A spherical iron ball 8 in. in diameter is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 10 in³/min, how fast is the thickness of the ice decreasing when it is 2 in. thick? How fast is the outer surface area of ice decreasing?