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
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 3.3.6
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x³/3 + x²/2 + x/4
Verified step by step guidance1
Step 1: Identify the function for which you need to find the derivatives. The given function is \( y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4} \).
Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \).
Step 3: Differentiate each term separately: \( \frac{d}{dx} \left( \frac{x^3}{3} \right) = x^2 \), \( \frac{d}{dx} \left( \frac{x^2}{2} \right) = x \), and \( \frac{d}{dx} \left( \frac{x}{4} \right) = \frac{1}{4} \). Combine these results to get the first derivative.
Step 4: To find the second derivative, differentiate the first derivative. Apply the power rule again to each term of the first derivative.
Step 5: Differentiate each term of the first derivative: \( \frac{d}{dx} (x^2) = 2x \), \( \frac{d}{dx} (x) = 1 \), and \( \frac{d}{dx} \left( \frac{1}{4} \right) = 0 \). Combine these results to get the second derivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Rule for Derivatives
The power rule is a basic principle in calculus used to find the derivative of a function of the form f(x) = x^n. The rule states that the derivative, f'(x), is n*x^(n-1). This rule simplifies the process of differentiation, allowing us to easily find the rate of change of polynomial functions.
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Power Rules
Sum Rule for Derivatives
The sum rule for derivatives states that the derivative of a sum of functions is the sum of their derivatives. If you have a function y = f(x) + g(x), the derivative y' is f'(x) + g'(x). This rule allows us to differentiate each term in a polynomial separately and then combine the results.
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Second Derivative
The second derivative of a function is the derivative of the first derivative, providing information about the curvature or concavity of the original function. It is denoted as f''(x) or d²y/dx². Calculating the second derivative helps in understanding the acceleration or the rate of change of the rate of change of the function.
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Related Practice
Textbook Question
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?
Textbook Question
If y = x² and dx/dt = 3, then what is dy/dt when x = –1?
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?