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If $ f(x) = x^2 + 10 \sin x $, show that there is a number $ c $ such that $ f(c) = 1000 $.

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04:51

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Calvin B.

August 10, 2022

I put this into my calc and it looks wayyyyyyy different

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

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.

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All right, we have a function F of x equals x squared plus 10 times sine of X. And we want to show that there is some numbers. See where uh FFC will equal 1000. And so I have this function graft on the desk. Most graphing calculator. Here it is. So here's the great for the function expert plus 10 times sine of X. Now, what I want you to notice is that okay? Well, here is a point on the graph. There's some X move that back up. There is uh some just looking worried. Okay, here here's your horizontal X axis right here. I don't care if I don't touch this move the graph here is your horizontal X axis right here. Now uh here is where it looks like The function would equal 1000. So there's some point right here. Okay, approximately here where the function equals 1000. So there is some corresponding X value down here on the X axis where the function equals 1000. But to be more formal uh more mathematically correct as to why? Uh there is a number, see down here on the X axis where the function does have a value of 1000. Uh This X square plus 10 times sine of X is a continuous function. Clearly for these x values here, The function is less than 1000. And clearly for X values here to function is greater than 1000. So because F of X is a continuous function and there are X values where the function is less than 1000 and X values. Where it's greater than 1000 then there must be some X value in between those other X values where the function is equal to exactly 1000. So because f of X is less than 1000, and f of x elsewhere is greater than 1000. And most importantly, because f of X is a continuous function, uh there will be a point a number see, on the X axis where the function at See will equal 1000.

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