Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 10

Chapter 4, Problem 10

Evaluate lim_x→2 (x³ - 3x² + 2) / (x-2) using l’Hôpital’s Rule and then check your work by evaluating the limit using an appropriate Chapter 2 method.

Verified step by step guidance

First, identify that the limit lim_x→2 (x³ - 3x² + 2) / (x-2) is an indeterminate form of type 0/0, which is suitable for l'Hôpital's Rule.

Apply l'Hôpital's Rule, which states that if lim_x→c f(x)/g(x) is of the form 0/0 or ∞/∞, then lim_x→c f(x)/g(x) = lim_x→c f'(x)/g'(x), provided the limit on the right exists.

Differentiate the numerator f(x) = x³ - 3x² + 2 to get f'(x) = 3x² - 6x.

Differentiate the denominator g(x) = x - 2 to get g'(x) = 1.

Evaluate the limit lim_x→2 (3x² - 6x) / 1, which simplifies to lim_x→2 (3x² - 6x). Now, check your work by factoring the original expression: (x³ - 3x² + 2) = (x-2)(x² - x - 1) and cancel the (x-2) terms, then evaluate lim_x→2 (x² - x - 1).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating 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 as x approaches a value, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In the given limit, the expression x³ - 3x² + 2 is a polynomial, and understanding its behavior as x approaches a specific value is crucial for limit evaluation, especially when factoring or simplifying the expression.

Limit Evaluation Techniques

Limit evaluation techniques include various methods for finding the limit of a function as it approaches a certain point. These methods can involve direct substitution, factoring, rationalizing, or using L'Hôpital's Rule. In this case, after applying L'Hôpital's Rule, one can also check the limit by substituting the value directly into the simplified expression or by using polynomial long division.

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Textbook Question

For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.

a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .

b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.

ƒ(x) = x⁵/16 ; [-2, 2] <IMAGE>

Textbook Question

Use ƒ' and ƒ" to complete parts (a) and (b).

a. Find the intervals on which f is increasing and the intervals on which it is decreasing.

b. Find the intervals on which f is concave up and the intervals on which it is concave down.

ƒ(x) = x⁹/9 + 3x⁵ - 16x

Textbook Question

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

Plot a possible graph of f.

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Textbook Question

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.

g. Use your work in parts (a) through (f) to sketch a graph of ƒ .

Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ -1 (x⁴ + x³ + 2x + 2) / (x + 1)

Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→2 (x² - 2x / (x² - 6x + 8)