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If $y=f(x)=2 x^{2}+2 x+3,$ find(a) $d y;$(b) $\Delta y;$(c) $\epsilon;$(d) Evaluate each of these quantities if:(i) $x=3$ and $\Delta x=1;$(ii) $x=3$ and $\Delta x=0.01.$

(a) $d y=(4 x+2) d x$(b) $(4 x+2) \Delta x+2(\Delta x)^{2}$(c) $2 \Delta x$(d) (i) 14,16,2(ii) 0.14,0.1402,0.02

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Oregon State University

Harvey Mudd College

Baylor University

Boston College

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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Find the indicated quantit…

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Evaluate $d y$ and $\Delta…

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If $y=x^{2}-3,$ find the v…

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01:52

Evaluate each function.

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01:45

So we have another problem we're looking at why equals three X. Squared. And our task is to find delta Y. Over delta X. That's a that's what we need to do to finish this problem. Um And what that means is we have to do the change in why that's what delta means. You subtract the values X two minus X. One. Now they actually already give you the values for X. One and Y. Sorry X. One and X two in the directions. They tell you that X one is 2, the next two is three. So in my equation down here I'm going to replace X. Two with three because that's what it equals And X. one or 2 because that's what that equals. What we don't know is what why one or why two are equal to. But we have the equation that we can plug in. I'm gonna start with X 23 squared is nine Times three would give me 27. I'm starting with that one because I wrote it first. That's my only reason. So now when I plug in to the two squared is four Times three would give me 12. Subtract those values. Uh, 27 -12 gives me 15 over one. So my answer here would be 15 Delta Wye, River Delta X is equal to 15.

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