Textbook Question
In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
y = (1 / x³), (−2, −1/8)
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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
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x⁵ - 0.125x² + 0.25x
Verified step by step guidance1
Identify the function for which you need to find the derivative: 𝔂 = x⁵ - 0.125x² + 0.25x.
Apply the power rule for differentiation, which states that the derivative of xⁿ is n*xⁿ⁻¹. Start with the first term, x⁵. The derivative is 5*x⁴.
Move to the second term, -0.125x². Using the power rule, the derivative is -0.125 * 2 * x¹, which simplifies to -0.25x.
Differentiate the third term, 0.25x. The derivative of x is 1, so the derivative of 0.25x is 0.25.
Combine the derivatives of each term to find the derivative of the entire function: 𝔂' = 5x⁴ - 0.25x + 0.25.
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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 how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In practical terms, the derivative provides the slope of the tangent line to the curve of the function at any given point.
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Derivatives
Power Rule
The power rule is a fundamental technique for finding derivatives of polynomial functions. It states that if a function is in the form f(x) = x^n, where n is a real number, then its derivative f'(x) is given by f'(x) = n*x^(n-1). This rule simplifies the differentiation process for terms involving powers of x.
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Sum Rule
The sum rule in calculus states that the derivative of a sum of functions is equal to the sum of their derivatives. If f(x) = g(x) + h(x), then the derivative f'(x) = g'(x) + h'(x). This rule allows for the differentiation of complex functions by breaking them down into simpler components.
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Related Practice
Textbook Question
Theory and Examples
The equations in Exercises 49 and 50 give the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = π/4 sec.
s = 2 − 2 sin t
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Textbook Question
For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.
lim (x → 1) (x⁵⁰ − 1) / (x − 1)
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Textbook Question
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the volume V = x³ of a cube when the edge lengths change from x₀ to x₀ + dx
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Textbook Question
Graphs
Match the functions graphed in Exercises 27–30 with the derivatives graphed in the accompanying figures (a)–(d).
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Textbook Question
Is there a value of c that will make
f(x) = { (sin²(3x)) / x², x ≠ 0
c, x = 0
continuous at x = 0? Give reasons for your answer.
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