Show that a linear function is increasing if and only if the slope of its graph is positive.

Name: Problem 47, Chapter 2: Precalculus A Prelude to Calculus
Uploaded: 2022-02-10T11:37:40-08:00
Duration: 3 min 23 s
Channel: Vaibhav Jain
Description: Show that a linear function is increasing if and only if the slope of its graph is positive.

Show that a linear function is increasing if and only if the...

Key Concepts

Linear Function

A linear function is a function of the form f(x) = mx + b, where m is the slope and b is the y-intercept. It represents a straight line on the coordinate plane and serves as a fundamental example of a polynomial function, playing a crucial role in understanding linear relationships, algebraic structures, and basic calculus concepts.

Slope

The slope in a linear function indicates the rate of change of the function, describing how much f(x) increases or decreases with a unit increase in x. It is a central concept in calculus and algebra, as it quantifies the steepness and direction of a line, and is key for understanding rates of change, derivatives, and linear approximations.

Increasing Function/Monotonicity

An increasing function is one where the output values rise as the input values rise. In the context of linear functions, the function is increasing if and only if the slope is positive. This concept is part of the broader study of function behavior (monotonicity), which is critical in calculus and mathematical analysis for understanding optimization, limits, and convergence.

Logical Equivalence (If and Only If)

The phrase ‘if and only if’ establishes a biconditional relationship between two statements, meaning that each statement implies the other. In mathematics, this logical structure is important for proving statements rigorously, ensuring that a condition is both necessary and sufficient for a particular property to hold, as seen in the characterization of increasing linear functions by a positive slope.

Transcript

00:01 Hello students, we are given a question, there is a function f, which is linear function, which has a positive slope.

00:05 We need to show that the average rate of change of a function in the particular interval, a comma b, is always positive, and we need to show again that f is always increasing function.

00:16 First of all, we are supposed to know that there is a theorem which can be said as lagrelanjee theorem.

00:20 We can apply here also, and first of all, we will go with the average rate of change.

00:25 See, what is average rate of change? average rate of change it is nothing but the difference between the function the function value okay with respect to a particular x points x points divided by the difference between the points okay we can write it as the form of y2 minus y1 divided by x2 minus x1 now if we can evaluate it we can write it in another form just like suppose function is already given f and is a layer of function so basically it's a derivative will be f dash x okay so basically it can be written as y2 minus y1 divided by x2 minus x1 and suppose there is a value x there is a value there is a particular value c okay and which belongs to a comma b it means there is a value some some value of x which is c and it exists in the particular domain particular particular interval which is a comma b okay so we can say as c belongs to a comma b so suppose there is a value of x equal to c now we can put it as f c is equal to okay now y2 is nothing but f of b we can write we can write it as f of b minus f of a by by 1 can be written as f of a sorry divided by here a or a and b and also x values so we can write it as b minus a here what we are given that the slope is positive see f dash c is nothing but the slope of a function at a particular point c which exists in the particular interval a comma b.

02:00 So we are given that f -tash six greater than zero positive slope.

02:04 Okay, so we can write that f of b minus f of a divided by b minus a always greater than zero.

02:13 Okay, students.

02:14 So here we can we are observed we can observe one thing as well that the we need to show that the average rate of change in the f interval a comma b is positive...

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