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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 24
Capitolo 3, Problema 24

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

Guida verificata passo dopo passo
1
First, identify the function f(x) given in the problem, which is f(x) = x² − 3x + 4.
Next, substitute f(x) into the alternative formula for derivatives: f'(x) = lim (z → x) (f(z) − f(x)) / (z − x).
Calculate f(z) by substituting z into the function: f(z) = z² − 3z + 4.
Substitute f(z) and f(x) into the formula: f'(x) = lim (z → x) ((z² − 3z + 4) − (x² − 3x + 4)) / (z − x).
Simplify the expression inside the limit: f'(x) = lim (z → x) ((z² − x²) − 3(z − x)) / (z − x). Factor the numerator and evaluate the limit as z approaches x.

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Derivative

A derivative represents the rate at which a function is changing at any given point and is a fundamental concept in calculus. It is the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, the derivative of f(x) = x² − 3x + 4 will be calculated using the alternative formula for derivatives.
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Percorso guidato
05:44
Derivatives

Limit

The limit is a fundamental concept in calculus that describes the value that a function approaches as the input approaches a certain point. In the alternative formula for derivatives, the limit is used to find the instantaneous rate of change by considering the behavior of the function as z approaches x. Understanding limits is crucial for applying this formula effectively.
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05:50
One-Sided Limits

Alternative Formula for Derivatives

The alternative formula for derivatives, f'(x) = lim (z → x) (f(z) − f(x)) / (z − x), provides a method to calculate the derivative by considering the limit of the difference quotient as z approaches x. This formula is particularly useful for understanding the derivative conceptually, as it directly relates to the definition of the derivative as the slope of the tangent line at a point.
Video consigliato:
Percorso guidato
05:44
Derivatives
Pratica correlata
Domanda del libro di testo

Economics


Marginal revenue

Suppose that the revenue from selling x washing machines is


r(x) = 20000(1 − 1/x) dollars.


b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week.

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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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Domanda del libro di testo

[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.

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Domanda del libro di testo

[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?

235
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Domanda del libro di testo

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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