Domanda del libro di testo
2. Express the following logarithms in terms of ln 5 and ln 7.
d. ln 1225
14
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.1.70d
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2
Guida verificata passo dopo passo1
Identify the given function as \( y = f(x) = \frac{x^3}{x^2 + 1} \) and the point \( x_0 = \frac{1}{2} \). The function \( g \) is the inverse of \( f \), so \( g = f^{-1} \). The point on \( g \) corresponding to \( x_0 \) is \( (f(x_0), x_0) \).
Calculate \( f(x_0) \) by substituting \( x_0 = \frac{1}{2} \) into \( f(x) \):
\[
f\left(\frac{1}{2}\right) = \frac{\left(\frac{1}{2}\right)^3}{\left(\frac{1}{2}\right)^2 + 1}
\]
Find the derivative \( f'(x) \) using the quotient rule:
\[
f'(x) = \frac{(3x^2)(x^2 + 1) - x^3(2x)}{(x^2 + 1)^2}
\]
Then evaluate \( f'(x_0) \) at \( x_0 = \frac{1}{2} \).
Use Theorem 1, which states that the derivative of the inverse function at \( y_0 = f(x_0) \) is the reciprocal of the derivative of the original function at \( x_0 \):
\[
g'(y_0) = \frac{1}{f'(x_0)}
\]
This gives the slope of the tangent line to \( g \) at the point \( (f(x_0), x_0) \).
Write the equation of the tangent line to \( g \) at \( (f(x_0), x_0) \) using point-slope form:
\[
y - x_0 = g'(f(x_0)) (x - f(x_0))
\]
where \( y \) and \( x \) are the coordinates on the graph of \( g \).
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:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Inverse Functions and Their Graphs
An inverse function reverses the roles of inputs and outputs of the original function, swapping x and y. Graphically, the inverse function's graph is the reflection of the original function's graph across the line y = x (the 45° line). Understanding this symmetry is crucial for locating points and tangent lines related to inverse functions.
Video consigliato:
3:17
Inverse Tangent
Derivative of an Inverse Function (Theorem 1)
Theorem 1 states that if a function f is differentiable and invertible at x₀, then the derivative of its inverse g at y₀ = f(x₀) is given by g'(y₀) = 1 / f'(x₀). This relationship allows us to find the slope of the tangent line to the inverse function using the derivative of the original function.
Video consigliato:
06:35Derivatives of Other Inverse Trigonometric Functions
Tangent Line Equation and Point-Slope Form
The tangent line to a curve at a point is the best linear approximation of the function near that point. Its equation can be found using the point-slope form: y - y₁ = m(x - x₁), where m is the slope (derivative at the point) and (x₁, y₁) is the point of tangency. This is essential for writing the tangent line to g at (f(x₀), x₀).
Video consigliato:
Percorso guidato
05:13Slopes of Tangent Lines
Pratica correlata
Domanda del libro di testo
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2
22
views
Domanda del libro di testo
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
e. Plot the functions f and g, the identity, the two tangent lines, and the line segment joining the points (x_0, f(x_0)) and (f(x_0), x_0). Discuss the symmetries you see across the main diagonal y=x.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
27
views
Domanda del libro di testo
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
25
views
Domanda del libro di testo
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
17
views
Domanda del libro di testo
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
e. x - 2ln(x)
19
views