Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 2.5.72

Capitolo 2, Problema 2.5.72

Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.
x³ − 15x + 1 = 0 (three roots)

Guida verificata passo dopo passo

Understand the Intermediate Value Theorem (IVT): It states that if a continuous function f(x) takes on values of opposite signs at two points a and b, then there exists at least one c in the interval (a, b) such that f(c) = 0.

Identify the function: Here, the function is f(x) = x³ − 15x + 1. This is a polynomial function, which is continuous everywhere.

Choose appropriate intervals: To apply the IVT, we need to find intervals where the function changes sign. Start by evaluating f(x) at various points to find intervals where f(x) changes from positive to negative or vice versa.

Evaluate the function at chosen points: Calculate f(x) at several points, such as f(-3), f(0), f(3), etc., to find intervals where the sign changes. For example, if f(-3) < 0 and f(0) > 0, then there is a root in the interval (-3, 0). Repeat this process to find all intervals where the function changes sign.

Use a graphing calculator or computer grapher: Once you have identified the intervals using the IVT, use a graphing tool to visualize the function and find the approximate values of the roots within those intervals.

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Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different signs at the endpoints (f(a) and f(b)), then there exists at least one c in (a, b) such that f(c) = 0. This theorem is fundamental for proving the existence of roots in continuous functions.

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Fundamental Theorem of Calculus Part 1

Continuous Functions

A continuous function is one where small changes in the input result in small changes in the output. Formally, a function f is continuous at a point x = c if the limit of f as x approaches c equals f(c). Understanding continuity is crucial for applying the Intermediate Value Theorem effectively.

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Graphing Techniques

Graphing techniques involve using tools like graphing calculators or software to visualize functions. By plotting the function, one can identify where it crosses the x-axis, indicating the roots of the equation. This visual approach complements analytical methods like the Intermediate Value Theorem in finding solutions.

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Pratica correlata

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Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

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Using the Sandwich Theorem

a. It can be shown that the inequalities 1 − x²/ 6 < (x sin x) / (2−2cos x) < 1 hold for all values of x close to zero (except for x = 0). What, if anything, does this tell you about limx→0 (x sin x) / (2 − 2cos x)?

Give reasons for your answer.

[Technology Exercise] b. Graph y = 1 − (x²/6), y=(x sinx)/(2 − 2cos x), and y = 1 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

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Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x² sin (1/x) = 0

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Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim ϴ → 0 cos (πϴ/sin ϴ)