Let 1\end{cases}"\u003eg(x)={5x−2if x1g\left(x\right)=\begin{cases}5x-2 & \text{if }x1\end{cases}g(x)=⎩⎨⎧5x−2aax2+bxif x1.Determine values of the constants aa and bb , if possible, for which gg is continuous at x=1x=1.

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

문제 2.R.79

2장, 문제 2.R.79

Let $g$\left$(x$\right$)=$\begin{cases}$5x-2 & $\text{if }$x<1\\ a & $\text{if }$x=1\\ ax^2+bx & $\text{if }$x>1$\end{cases}$$ .

Determine values of the constants $a$ and $b$ , if possible, for which $g$ is continuous at $x equals 1$ .

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To ensure the function g(x) is continuous at x = 1, the left-hand limit, right-hand limit, and the value of the function at x = 1 must all be equal.

First, calculate the left-hand limit as x approaches 1. For x < 1, g(x) = 5x - 2. Thus, the left-hand limit is lim_{x $\to$ 1^-} g(x) = 5(1) - 2 = 3.

Next, calculate the right-hand limit as x approaches 1. For x > 1, g(x) = ax^2 + bx. Thus, the right-hand limit is lim_{x $\to$ 1^+} g(x) = a(1)^2 + b(1) = a + b.

The function value at x = 1 is given by g(1) = a.

For g(x) to be continuous at x = 1, set the left-hand limit equal to the right-hand limit and the function value: 3 = a and 3 = a + b. Solve these equations to find the values of a and b.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function g(x) has three distinct cases depending on whether x is less than, equal to, or greater than 1. Understanding how to evaluate and analyze piecewise functions is crucial for determining continuity and limits at specific points.

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 g(x) to be continuous at x=1, the left-hand limit (as x approaches 1 from the left) must equal the right-hand limit (as x approaches 1 from the right) and also equal g(1). This concept is essential for solving the problem of finding appropriate values for a and b.

Limits

Limits describe the behavior of a function as it approaches a certain point. In this context, we need to calculate the limits of g(x) as x approaches 1 from both sides. The values of a and b can be determined by ensuring that these limits match the value of g(1), which is defined as a, thus ensuring the function is continuous at that point.