Table of contents

0. Functions7h 52m
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 Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Properties of Functions

Calculus

0. Functions

Properties of Functions

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1:55 minutes

Problem 34d

Textbook Question

Find the largest interval on which the given function is increasing.

d. R(x) = √ 2x - 1

Verified step by step guidance

First, understand that a function is increasing on an interval if its derivative is positive on that interval. So, we need to find the derivative of the function R(x) = √(2x - 1).

To find the derivative, use the chain rule. The function can be rewritten as (2x - 1)^(1/2). The derivative of this function is (1/2)(2x - 1)^(-1/2) * (d/dx)(2x - 1).

Calculate the derivative of the inner function (2x - 1), which is simply 2. Therefore, the derivative of R(x) is (1/2)(2x - 1)^(-1/2) * 2.

Simplify the expression for the derivative: R'(x) = (2/2)(2x - 1)^(-1/2) = (2x - 1)^(-1/2).

Determine where the derivative is positive. Since (2x - 1)^(-1/2) is positive when 2x - 1 > 0, solve the inequality 2x - 1 > 0 to find the interval where the function is increasing.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. To determine where a function is increasing, we analyze its derivative: if the derivative is positive over an interval, the function is increasing on that interval.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are essential for identifying intervals of increase or decrease, as they can indicate potential local maxima or minima, which help in determining the overall behavior of the function.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It is crucial for expressing the intervals on which a function is increasing or decreasing. For example, the interval (a, b) indicates that the function is increasing from point a to point b, not including the endpoints.