Skip to main content
Ch. 3 - Derivatives
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. 3 - DerivativesProblema 94a
Capitolo 3, Problema 94a

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.
a. g(1)

Guida verificata passo dopo passo
1
Step 1: Understand the problem. We are given a function g(x) = f(x^2) and need to find g(1). We know that f is differentiable and passes through the point (1, 4), and the tangent line to f at (1, 4) is y = 3x + 1.
Step 2: Evaluate g(1). Since g(x) = f(x^2), we substitute x = 1 into g(x) to get g(1) = f(1^2) = f(1).
Step 3: Use the information about f. We know that the graph of f passes through the point (1, 4), which means f(1) = 4.
Step 4: Conclude the evaluation. Since f(1) = 4, we have g(1) = f(1) = 4.
Step 5: Summarize the result. Therefore, g(1) is equal to 4.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
1m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Differentiable Functions

A differentiable function is one that has a derivative at every point in its domain. This means that the function is smooth and continuous, allowing for the calculation of slopes of tangent lines at any point. The existence of a derivative indicates that the function can be locally approximated by a linear function, which is essential for understanding tangent lines.
Video consigliato:
Percorso guidato
05:53
Finding Differentials

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. The equation of the tangent line can be derived using the point-slope form, which incorporates the derivative of the function at that point. In this case, the tangent line to f at (1, 4) is given as y = 3x + 1, indicating that the slope at that point is 3.
Video consigliato:
Percorso guidato
05:13
Slopes of Tangent Lines

Composition of Functions

The composition of functions involves combining two functions where the output of one function becomes the input of another. In this problem, g(x) = f(x²) represents a composition where the input x is squared before being passed to function f. Understanding how to evaluate g(1) requires substituting 1 into the composition and then determining the value of f at the resulting input.
Video consigliato:
3:48
Evaluate Composite Functions - Special Cases
Pratica correlata
Domanda del libro di testo

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.

c. g'(π)

363
views
Domanda del libro di testo

Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

y = f(x)g(x)

287
views
Domanda del libro di testo

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.


a. Find L′(1) and interpret the meaning of this value.

278
views
Domanda del libro di testo

Derivatives from tangent lines Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

b. y = f(x) / g(x)

289
views
Domanda del libro di testo

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

281
views
Domanda del libro di testo

Composition containing sin x Suppose f is differentiable for all real numbers with f(0)=−3,f(1)=3,f′(0)=3, and f′(1)=5. Let g(x)=sin(πf(x)). Evaluate the following expressions.

b. g'(1)

357
views