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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 25a
Capitolo 3, Problema 25a

[Technology Exercise]


Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula


y = 6(1 - t/12)² m.


a. Find the rate dy/dt (m/h) at which the tank is draining at time t.

Guida verificata passo dopo passo
1
Identify the given function for the depth of fluid in the tank: \( y = 6(1 - \frac{t}{12})^2 \). This function represents the depth of the fluid in meters as a function of time \( t \) in hours.
To find the rate at which the tank is draining, we need to compute the derivative of \( y \) with respect to \( t \), which is \( \frac{dy}{dt} \). This derivative will give us the rate of change of the fluid depth over time.
Apply the chain rule to differentiate \( y = 6(1 - \frac{t}{12})^2 \). The chain rule states that if you have a composite function \( f(g(t)) \), then \( \frac{d}{dt}[f(g(t))] = f'(g(t)) \cdot g'(t) \).
First, differentiate the outer function \( f(u) = 6u^2 \) with respect to \( u \), where \( u = 1 - \frac{t}{12} \). The derivative is \( f'(u) = 12u \).
Next, differentiate the inner function \( g(t) = 1 - \frac{t}{12} \) with respect to \( t \). The derivative is \( g'(t) = -\frac{1}{12} \). Combine these using the chain rule: \( \frac{dy}{dt} = 12(1 - \frac{t}{12}) \cdot (-\frac{1}{12}) \).

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Derivative

The derivative of a function represents the rate at which the function's value changes with respect to a change in its input. In this context, dy/dt is the derivative of the depth y with respect to time t, indicating how quickly the fluid level in the tank is decreasing over time.
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Percorso guidato
05:44
Derivatives

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This rule is essential for finding dy/dt when y is expressed as a function of t.
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05:02
Intro to the Chain Rule

Quadratic Function

A quadratic function is a polynomial function of degree two, typically in the form ax² + bx + c. In this problem, y = 6(1 - t/12)² is a quadratic function of t. Understanding its structure helps in applying the chain rule and finding the derivative, as it involves recognizing the inner function (1 - t/12) and its square.
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6:04
Introduction to Polynomial Functions
Pratica correlata
Domanda del libro di testo

The number of gallons of water in a tank t minutes after the tank has started to drain is Q(t) = 200(30 - t)². How fast is the water running out at the end of 10 min? What is the average rate at which the water flows out during the first 10 min?

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Economics


Marginal revenue

Suppose that the revenue from selling x washing machines is


r(x) = 20000(1 − 1/x) dollars.


c. Find the limit of r'(x) as x → ∞. How would you interpret this number?

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[Technology Exercise]


Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula


y = 6(1 - t/12)² m.


b. When is the fluid level in the tank falling fastest? Slowest? What are the values of dy/dt at these times?

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Domanda del libro di testo

Vehicular stopping distance Based on data from the U.S. Bureau of Public Roads, a model for the total stopping distance of a moving car in terms of its speed is s = 1.1v + 0.054v², where s is measured in ft and v in mph. The linear term 1.1v models the distance the car travels during the time the driver perceives a need to stop until the brakes are applied, and the quadratic term 0.054v² models the additional braking distance once they are applied. Find ds/dv at v = 35 and v = 70 mph, and interpret the meaning of the derivative.

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Using the Alternative Formula for Derivatives


Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.


f(x) = x² − 3x + 4

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Additional Applications


Bacterium population

When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 10⁶ + 10⁴t − 10³t². Find the growth rates at


a. t = 0 hours.

b. t = 5 hours.

c. t = 10 hours.

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