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Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.7.22
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - 1) / (x² + 3x)
Guida verificata passo dopo passo1
First, substitute x = 0 into the limit expression to check if it results in an indeterminate form. Substituting gives (e^0 - 1) / (0^2 + 3*0) = 0/0, which is an indeterminate form.
Since the limit is in the indeterminate form 0/0, we can apply l'Hôpital's Rule. This rule states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then the limit is the same as the limit of f'(x)/g'(x) as x approaches that value, provided the derivatives exist.
Differentiate the numerator and the denominator separately. The derivative of the numerator, e^x - 1, is e^x. The derivative of the denominator, x² + 3x, is 2x + 3.
Apply l'Hôpital's Rule by taking the limit of the new expression: lim_x→0 (e^x) / (2x + 3).
Substitute x = 0 into the new expression to evaluate the limit: (e^0) / (2*0 + 3) = 1/3. Thus, the limit is 1/3.
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Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for evaluating expressions that may be indeterminate, such as 0/0 or ∞/∞. In this question, we need to analyze the behavior of the function as x approaches 0 to determine the limit.
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One-Sided Limits
l'Hôpital's Rule
l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This technique simplifies the evaluation of complex limits, making it particularly useful in this problem.
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5:50Power Rules
Exponential Functions
Exponential functions, such as eˣ, are functions where a constant base is raised to a variable exponent. They exhibit unique properties, including the fact that eˣ approaches 1 as x approaches 0. Understanding the behavior of exponential functions is essential for evaluating limits involving them, as they often lead to indeterminate forms that require careful analysis.
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Trajectory high point A stone is launched vertically upward from a cliff 192 ft above the ground at a speed of 64 ft/s. Its height above the ground t seconds after the launch is given by s = -16t² + 64t + 192, for 0 ≤ t ≤ 6. When does the stone reach its maximum height?
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23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3x + 1) (4 - x) dx
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{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.
f(x) = tan x - 2x; x₀ = 1.2
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