Textbook Question
Derivatives Find and simplify the derivative of the following functions.
h(x) = (x−1)(2x²-1) / (x³-1)
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.5.66
Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
lim x→π/2 cos x/x−π/2
Verified step by step guidance1
Recognize that the given limit can be interpreted as the derivative of a function at a specific point. The expression lim x→π/2 (cos x)/(x−π/2) resembles the definition of a derivative.
Recall the definition of the derivative of a function f at a point a: f'(a) = lim x→a (f(x) - f(a))/(x - a).
Identify the function f(x) = cos x and the point a = π/2. Notice that f(a) = cos(π/2) = 0.
Rewrite the limit in the form of the derivative definition: lim x→π/2 (cos x - cos(π/2))/(x - π/2). This matches the derivative definition for f(x) = cos x at x = π/2.
Conclude that the limit represents the derivative of f(x) = cos x at x = π/2, which is f'(π/2). To find this derivative, compute f'(x) = -sin x and evaluate it at x = π/2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
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Derivatives
The derivative of a function at a point measures the rate at which the function's value changes as its input 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 this context, recognizing the limit as a derivative allows for the application of derivative rules to evaluate the limit.
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L'Hôpital's Rule
L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule is particularly useful in simplifying complex limit problems.
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Related Practice
Textbook Question
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(sec⁴x tan² x)
Textbook Question
A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.
Textbook Question
Find y'' for the following functions.
y = cos θ sin θ
Textbook Question
Evaluate the derivative of the following functions.
f(x) = sin-1 (e-2x)
Textbook Question
Derivatives Find and simplify the derivative of the following functions.
f(t) = t⁵/³e^t