A personal-computer salesperson receives a base salary of ...

A personal-computer salesperson receives a base salary of $\$ 1,000$ per month and a commission of $5 \%$ of all sales over $\$ 10,000$ during the month. If the monthly sales are $\$ 20,000$ or more, then the salesperson is given an additional $\$ 500$ bonus. Let $E(s)$ represent the person's earnings per month as a function of the monthly sales $s$. (A) Graph $E(s)$ for $0 \leq s \leq 30,000$. (B) Find $\lim _{s \rightarrow 10,000} E(s)$ and $E(10,000)$. (C) Find $\lim _{s \rightarrow 20,000} E(s)$ and $E(20,000)$. (D) Is $E$ continuous at $s=10,000$ ? At $s=20,000 ?$

A personal-computer salesperson receives a base salary of $\$ 1,000$ per month and a commission of $5 \%$ of all sales over $\$ 10,000$ during the month. If the monthly sales are $\$ 20,000$ or more, then the salesperson is given an additional $\$ 500$ bonus. Let $E(s)$ represent the person's earnings per month as a function of the monthly sales $s$.
(A) Graph $E(s)$ for $0 \leq s \leq 30,000$.
(B) Find $\lim _{s \rightarrow 10,000} E(s)$ and $E(10,000)$.
(C) Find $\lim _{s \rightarrow 20,000} E(s)$ and $E(20,000)$.
(D) Is $E$ continuous at $s=10,000$ ? At $s=20,000 ?$

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Key Concepts

Piecewise Functions

A piecewise function is defined using different expressions depending on the input's value in specific intervals. These functions are useful in modeling scenarios where the behavior or rule governing the variable changes at certain thresholds.

Graphing Piecewise Functions

Graphing a piecewise function involves plotting each separate section over its corresponding interval. Care must be taken to correctly represent open and closed endpoints to accurately reflect the function’s definition across its different domains.

Limits of a Function

The limit of a function as the input approaches a certain value is a fundamental concept that describes the behavior of the function near that point. It is useful for understanding how a function behaves at points where its definition might change.

Continuity of a Function

A function is continuous at a point if the limit from both sides equals the function's actual value at that point, meaning there are no breaks, jumps, or holes. In piecewise functions, checking continuity involves verifying that the limits and the function's defined value coincide at the transition points.

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