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 3.9.15a
Faster than a calculator Use the approximation (1 + x)ᵏ ≈ 1 + kx to estimate the following.
a. (1.0002)⁵⁰
Verified step by step guidance1
Identify the expression to approximate: (1.0002)⁵⁰. Here, we can see that x = 0.0002 and k = 50.
Recognize that the approximation (1 + x)ᵏ ≈ 1 + kx is useful when x is small, which is the case here.
Substitute the values of x and k into the approximation formula: 1 + kx = 1 + 50 * 0.0002.
Calculate the product kx: 50 * 0.0002 = 0.01.
Add the result to 1 to complete the approximation: 1 + 0.01.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Approximation
The binomial approximation (1 + x)ᵏ ≈ 1 + kx is a simplification used when x is small and k is a constant. It allows for quick estimates of expressions raised to a power without complex calculations. This approximation is derived from the binomial theorem and is particularly useful for small perturbations around 1.
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Small x Assumption
The small x assumption is crucial for the validity of the binomial approximation. It implies that x is close to zero, making higher-order terms in the binomial expansion negligible. This assumption simplifies calculations and is often used in physics and engineering to approximate values efficiently.
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Exponentiation
Exponentiation is the mathematical operation involving numbers raised to a power, denoted as (1 + x)ᵏ. Understanding how to manipulate and approximate powers is essential in calculus, especially when dealing with series expansions and approximations. It forms the basis for many calculus concepts, including derivatives and integrals.
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Related Practice
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
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).
Textbook Question
In Exercises 47 and 48, find an equation for
(b) the horizontal tangent line to the curve at Q.
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Textbook Question
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?