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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.5.85
Chapter 3, Problem 3.5.85

Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?

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First, understand the concept of continuity for a function. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.
Identify the piecewise function g(x) given in the problem. Since the problem does not specify the pieces, assume g(x) is defined differently on different intervals. For example, g(x) might be defined as g(x) = x^2 for x < a and g(x) = 2x + 1 for x ≥ a.
To determine the values of 'a' for which g(x) is continuous, ensure that the left-hand limit and right-hand limit at x = a are equal, and also equal to g(a). This means you need to solve the equation: lim(x -> a^-) g(x) = lim(x -> a^+) g(x) = g(a).
Calculate the left-hand limit: lim(x -> a^-) g(x). If g(x) = x^2 for x < a, then this limit is a^2.
Calculate the right-hand limit: lim(x -> a^+) g(x). If g(x) = 2x + 1 for x ≥ a, then this limit is 2a + 1. Set the left-hand limit equal to the right-hand limit and solve for 'a': a^2 = 2a + 1.

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Key Concepts

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

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For piecewise functions, continuity must be checked at the boundaries where the pieces meet. This involves ensuring that the left-hand limit, right-hand limit, and the function's value at that point are all equal.
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Intro to Continuity

Piecewise Function

A piecewise function is defined by different expressions based on the input value. Each segment of the function applies to a specific interval of the domain. Understanding how to evaluate and analyze each piece is crucial for determining overall properties like continuity and differentiability.
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Piecewise Functions

Limit

The limit of a function describes the behavior of the function as it approaches a particular point from either side. In the context of continuity, limits are used to determine if the function approaches the same value from both directions at a point of interest. Evaluating limits is essential for confirming the continuity of piecewise functions at their transition points.
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One-Sided Limits
Related Practice
Textbook Question

Angle to a particle (part 2) The figure in Exercise 81 shows the particle traveling away from the sensor, which may have influenced your solution (we expect you used the inverse sine function). Suppose instead that the particle approaches the sensor (see figure). How would this change the solution? Explain the differences in the two answers. <IMAGE>

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Textbook Question

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.

h (x) = x^√x; a = 4

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Textbook Question

Consider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .

Textbook Question

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = tan¹⁰x / (5x+3)⁶

Textbook Question

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>


y = cx²; x²+2y² = k, where c and k are constants

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).


y = 4 log₃(x²−1)