Let 1\end{cases}"\u003eg(x)={x2+xif x1g\left(x\right)=\begin{cases}$$x^2+x$$ & \text{if }x1\end{cases}g(x)=⎩⎨⎧x2+xa3x+5if x1b. Determine the value of aa for which gg is continuous from the right at 11.

To determine the value of 'a' for which the function g(x) is continuous from the right at x = 1, we need to ensure that the right-hand limit of g(x) as x approaches 1 is equal to g(1). The right-hand limit of g(x) as x approaches 1 is found by considering the expression for g(x) when x \u003e 1, which is 3x + 5. Calculate the right-hand limit: $$ \lim_{{x \to 1^+}} g(x) = \lim_{{x \to 1^+}} (3x + 5) . $$Evaluate this limit by substituting x = 1 into the expression 3x + 5, which gives 3(1) + 5. For g(x) to be continuous from the right at x = 1, set the right-hand limit equal to g(1), which is 'a'. Therefore, solve the equation 3(1) + 5 = a to find the value of 'a'.

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.6.87b

Chapter 2, Problem 2.6.87b

Let $g$\left$(x$\right$)=$\begin{cases}$x^2+x & $\text{if }$x<1\\ a & $\text{if }$x=1\\ 3x+5 & $\text{if }$x>1$\end{cases}$$
b. Determine the value of $a$ for which $g$ is continuous from the right at $1$ .

Verified step by step guidance

To determine the value of 'a' for which the function g(x) is continuous from the right at x = 1, we need to ensure that the right-hand limit of g(x) as x approaches 1 is equal to g(1).

The right-hand limit of g(x) as x approaches 1 is found by considering the expression for g(x) when x > 1, which is 3x + 5.

Calculate the right-hand limit: $ \lim_{{x \to 1^+}} g(x) = \lim_{{x \to 1^+}} (3x + 5) $.

Evaluate this limit by substituting x = 1 into the expression 3x + 5, which gives 3(1) + 5.

For g(x) to be continuous from the right at x = 1, set the right-hand limit equal to g(1), which is 'a'. Therefore, solve the equation 3(1) + 5 = a to find the value of 'a'.

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Key Concepts

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

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 piecewise functions is crucial for determining their properties, such as continuity.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point from both sides equals the function's value at that point. For g(x) to be continuous at x=1, the limit as x approaches 1 from the left must equal the limit as x approaches 1 from the right, and both must equal g(1). This concept is essential for solving the problem.

Limits

Limits describe the behavior of a function as the input approaches a certain value. In this context, we need to find the left-hand limit (as x approaches 1 from values less than 1) and the right-hand limit (as x approaches 1 from values greater than 1) of g(x). Evaluating these limits will help determine the appropriate value of a that ensures continuity at x=1.

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Determine whether the following statements are true and give an explanation or counterexample.

The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

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The hyperbolic cosine function, denoted $$\cosh\]\left$(x$\right$)$ , is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $$\cosh\]\left$(x$\right$)=$\frac{e^{x}$+e^{-x}}{2}$ .

b. Evaluate $hyperbolic cosine of 0$ . Use symmetry and part (a) to sketch a plausible graph for $y = \cosh (x)$ .

Textbook Question

Assume you invest $250 at the end of each year for 10 years at an annual interest rate of $r$ . The amount of money in your account after 10 years is given by $A (r) = \frac{250 ({(1 + r)}^{10} - 1)}{r}$ . Assume your goal is to have $3500 in your account after 10 years.

b. Use a calculator to estimate the interest rate required to reach your financial goal.

Textbook Question

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1^+ f(x)

Textbook Question

Complete the following steps for the given functions.

b. Find the vertical asymptotes of f (if any).

$f (x) = \frac{3 x^{2} - 2 x + 5}{3 x + 4}$

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Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

Let g(x)={x2+xif x<1aif x=13x+5if x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)g(x)=⎩⎨⎧x2+xa3x+5if x<1if x=1if x>1b. Determine the value of aa for which gg is continuous from the right at 11.

Verified video answer for a similar problem:

Key Concepts

Piecewise Functions

Continuity

Limits

Let $g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)$
b. Determine the value of $a$ for which $g$ is continuous from the right at $1$ .