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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.RE.1a
Capitolo 7, Problema 7.RE.1a

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.

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1
Identify the given function: \(y = t + 1\).
Understand what it means for \(y\) to double when \(t\) increases by 1 unit. Doubling means the new value of \(y\) should be twice the original value.
Calculate the original value of \(y\) at some \(t\): \(y = t + 1\).
Calculate the new value of \(y\) when \(t\) increases by 1: \(y_{new} = (t + 1) + 1 = t + 2\).
Compare \(y_{new}\) to \$2y$: check if $t + 2 = 2(t + 1)$ holds for all $t$. If not, then $y$ does not double when $t$ increases by 1.

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Understanding Linear Functions

A linear function has the form y = mt + b, where m is the slope and b is the y-intercept. The slope m represents the rate of change of y with respect to t, meaning how much y changes when t increases by one unit.
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Linearization

Rate of Change vs. Doubling

Rate of change refers to the amount y increases or decreases per unit increase in t. Doubling means y becomes twice its previous value, which is a multiplicative change, not additive. Understanding this distinction is key to evaluating the statement.
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Intro To Related Rates

Counterexamples in Mathematical Reasoning

A counterexample disproves a general statement by providing a specific case where the statement fails. To test if y doubles when t increases by 1, substituting values can show whether the statement holds or not.
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