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(a) Suppose $f$ is continuous on $1 \leq x \leq 3,$ with $f(1)=7,$ and $f(3)=12$ Show that every $y$ -value between 7 and 12 is assumed at least once. (Hint:draw a sketch) (b) More generally, if $f$ is continuous on $a \leq x \leq b,$ with $f(a)=M,$ and $f(b)=N,$ then each $y$ -value between $M$ and $N$ is assumed at least once. (This is known as the Intermediate Value Theorem.)

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Baylor University

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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here. We want to show the function. Show the function is continuous for all values of X in the interval. And prove that f must have at least 10 in the interval. So here we're gonna be given aftereffects is equal to the text cube, my history expert. Mhm. Um -36 x plus 14. Mhm. What we see is that this is a polynomial so we know immediately that it is going to be continuous. There's no restrictions to it. So it's a continuous polynomial of degree three. Um And then we want to um Look show that it has at least 10. So we're going to use the intermediate value theorem here. Um Or Yeah. And then we know that a zero. So we're going to look between zero and one. Well, we know that F of zero. It's 14 And F of one is -23. So between zero and one we have this gap. Then we know that at some point it's a continuous function. It has to go through um zero point.

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