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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 64
Capitolo 1, Problema 64

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = 4x-3

Guida verificata passo dopo passo
1
Step 1: Start by substituting the function \( f(x) = 4x - 3 \) into the difference quotient formula \( \frac{f(x+h) - f(x)}{h} \).
Step 2: Calculate \( f(x+h) \) by replacing \( x \) with \( x+h \) in the function \( f(x) = 4x - 3 \). This gives \( f(x+h) = 4(x+h) - 3 \).
Step 3: Expand \( f(x+h) = 4(x+h) - 3 \) to get \( 4x + 4h - 3 \).
Step 4: Substitute \( f(x+h) = 4x + 4h - 3 \) and \( f(x) = 4x - 3 \) into the difference quotient: \( \frac{(4x + 4h - 3) - (4x - 3)}{h} \).
Step 5: Simplify the expression in the numerator: \( (4x + 4h - 3) - (4x - 3) = 4h \). The difference quotient becomes \( \frac{4h}{h} \).

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Difference Quotient

The difference quotient is a formula used to calculate the average rate of change of a function over an interval. It is expressed as (ƒ(x+h) - ƒ(x)) / h, where h represents a small change in x. This concept is fundamental in calculus as it leads to the definition of the derivative, which measures the instantaneous rate of change.
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The Quotient Rule

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this case, we evaluate the function ƒ(x) = 4x - 3 at two points: x and x + h. Understanding how to correctly substitute values into a function is crucial for simplifying expressions like the difference quotient.
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Percorso guidato
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Evaluating Composed Functions

Algebraic Simplification

Algebraic simplification is the process of manipulating an expression to make it easier to work with or to reveal its underlying structure. This includes combining like terms, factoring, and reducing fractions. In the context of the difference quotient, simplifying the expression after substituting the function values is essential to arrive at a clearer form, which can then be analyzed further.
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Determine Continuity Algebraically
Pratica correlata
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{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.


a. Is this function one-to-one on the interval 0 ≤ t ≤ 4?

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Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.


h(x)=−4x2−4x+12h(x)=-4x^{^2}-4x+12

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Domanda del libro di testo

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.


b. Find the inverse function that gives the time t at which the ball is at height h as the ball travels upward. Express your answer in the form t = ƒ⁻¹ (h)

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Domanda del libro di testo

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.


c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = ƒ⁻¹ (h)

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Domanda del libro di testo

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 10

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Domanda del libro di testo

Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.


(g o ƒ ) (x) = x²⸍³ + 3

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