Domanda del libro di testo
91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]
b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 7.6.4b
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
4. b. arcsin(-1/√2)
Guida verificata passo dopo passo1
Recall that the function \(\arcsin(x)\) gives the angle \(\theta\) whose sine is \(x\), with the range of \(\arcsin\) restricted to \([-\frac{\pi}{2}, \frac{\pi}{2}]\) (Quadrants IV and I).
Identify the value inside the \(\arcsin\): here it is \(-\frac{1}{\sqrt{2}}\). Recognize that \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), so the reference angle is \(\frac{\pi}{4}\).
Since the sine value is negative, and \(\arcsin\) outputs angles in \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the angle must be in Quadrant IV where sine is negative.
Use the reference triangle in Quadrant IV to express the angle as \(-\frac{\pi}{4}\), because sine of \(-\frac{\pi}{4}\) is \(-\frac{1}{\sqrt{2}}\).
Therefore, the angle \(\theta = \arcsin\left(-\frac{1}{\sqrt{2}}\right)\) corresponds to \(-\frac{\pi}{4}\) within the principal range of \(\arcsin\).
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Inverse Sine Function (arcsin)
The inverse sine function, arcsin, returns the angle whose sine value is a given number. Its range is limited to angles between -π/2 and π/2 (or -90° to 90°), meaning it outputs angles in the first and fourth quadrants. Understanding this helps identify the principal value of the angle.
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4:03
Inverse Sine
Reference Triangles
Reference triangles are right triangles used to find angle measures based on known trigonometric ratios. By considering the absolute value of the sine and the quadrant of the angle, you can determine the exact angle measure and its sign, aiding in solving inverse trig problems.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Quadrants and Sign of Trigonometric Functions
The sign of sine depends on the quadrant where the angle lies: sine is positive in the first and second quadrants and negative in the third and fourth. Knowing the quadrant helps determine the correct angle corresponding to a given sine value, especially when the value is negative.
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Percorso guidato
6:04Introduction to Trigonometric Functions
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:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2
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Domanda del libro di testo
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. √(x^4 + x^3)
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Domanda del libro di testo
3. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
b. ln(3x² - 9x) + ln(1/3x)
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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:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
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Domanda del libro di testo
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
2. y' = y²
b. y = -1/(x+3)
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