Domanda del libro di testo
Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).
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Ch. 3 - Derivatives
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 3, Problema 3.3.89
Calculator limits Use a calculator to approximate the following limits.
lim x🠂0 e^3x-1 / x
Guida verificata passo dopo passo1
Step 1: Recognize that the limit \( \lim_{x \to 0} \frac{e^{3x} - 1}{x} \) is an indeterminate form of type \( \frac{0}{0} \). This suggests that L'Hôpital's Rule might be applicable.
Step 2: Recall L'Hôpital's Rule, which states that if \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \), provided the limit on the right exists.
Step 3: Differentiate the numerator and the denominator separately. The derivative of the numerator \( e^{3x} - 1 \) is \( 3e^{3x} \), and the derivative of the denominator \( x \) is \( 1 \).
Step 4: Apply L'Hôpital's Rule to the original limit: \( \lim_{x \to 0} \frac{e^{3x} - 1}{x} = \lim_{x \to 0} \frac{3e^{3x}}{1} \).
Step 5: Evaluate the new limit: \( \lim_{x \to 0} 3e^{3x} \). As \( x \to 0 \), \( e^{3x} \to e^0 = 1 \). Therefore, the limit simplifies to \( 3 \times 1 = 3 \).
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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 of discontinuity or infinity. In this case, we are interested in the limit of the function as x approaches 0.
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One-Sided Limits
Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where e is the base of natural logarithms. They are characterized by their rapid growth or decay and are essential in various applications, including calculus. The expression e^(3x) in the limit indicates that we are dealing with an exponential function, which will influence the limit's value as x approaches 0.
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L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in this problem, as substituting x = 0 directly leads to an indeterminate form.
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Pratica correlata
Domanda del libro di testo
49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.
f (x) = (sin x)^In x; a = π/2
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Domanda del libro di testo
Find the derivative of the following functions.
y = cot x / (1 + csc x)
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Domanda del libro di testo
The function represents the position of an object at time t moving along a line. Suppose and . Find the average velocity of the object over the interval of time .
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Domanda del libro di testo
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (x+1)^3/2(x-4)^5/2 / (5x+3)^2/3
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Domanda del libro di testo
A ship leaves port traveling southwest at a rate of 12 mi/hr. At noon, the ship reaches its closest approach to a radar station, which is on the shore 1.5 mi from the port. If the ship maintains its speed and course, what is the rate of change of the tracking angle θ between the radar station and the ship at 1:30 P.M. (see figure)? (Hint: Use the Law of Sines.) <IMAGE>
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