Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
- 8. Definite Integrals3h 25m
0. Functions
Properties of Functions
Problem 34d
Textbook Question
Find the largest interval on which the given function is increasing.
d. R(x) = √ 2x - 1

1
To determine where the function \( R(x) = \sqrt{2x - 1} \) is increasing, we first need to find its derivative. The derivative will help us understand the behavior of the function.
Apply the chain rule to differentiate \( R(x) = \sqrt{2x - 1} \). The chain rule states that if you have a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \). Here, \( f(u) = \sqrt{u} \) and \( g(x) = 2x - 1 \).
The derivative of \( f(u) = \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \), and the derivative of \( g(x) = 2x - 1 \) is 2. Therefore, the derivative of \( R(x) \) is \( R'(x) = \frac{1}{2\sqrt{2x - 1}} \cdot 2 \).
Simplify the expression for the derivative: \( R'(x) = \frac{1}{\sqrt{2x - 1}} \). The function is increasing where its derivative is positive.
Since \( \frac{1}{\sqrt{2x - 1}} > 0 \) for \( 2x - 1 > 0 \), solve the inequality \( 2x - 1 > 0 \) to find the interval where the function is increasing. This gives \( x > \frac{1}{2} \). Therefore, the largest interval on which the function is increasing is \( (\frac{1}{2}, \infty) \).
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