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# DIFFERENTIAL CALCULUS MTH 251H

OSU

GPA 3.79

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This 3 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 251H at Oregon State University taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/224448/mth-251h-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15

Section 27 39I angents Velocities and Other Rates of Change Section 28 Derivatives All that is new in these sections is some convenient notation for expressing an instantaneous rate of change Recall our quest is to understand and effectively use the techniques of differ ential calculus Our mantra is Derivative 2 Rate of Change 2 Slope We already have progressed thus far if t is time and s s t is the position of an object moving on a straight path here the s axis at time t7 then the instantaneous rate of change of position With respect to time is the velocity 1 of the object 1 velocity at t rate of change of position With respect to time at t 7 the slope of the tangent line to the graph of f at the point t7f We also learned that the slope mT of the tangent line T to the curve by y f at the point af 1 is mT 7715 40 Where m5 is the slope of a secant line joining the two points afa and 7f Notice that f W i f a 5870 m5 is the average rate of change of the function f as f varies from a to L So mT limmna m5 is the instantaneous rate of change of the function f at 1 Brie y Slopes of secant lines are average rates of change Slopes of tangent lines are instantaneous rates of change in calculus a rate of change normally means an instantaneous rate of change Sometimes the word instantaneous is added for emphasis Let y f be any function of L We inquire about the rate of change of f at a As With the foregoing model problems involving velocity of slopes of tangents the limit of average rates of change of f Rate of change of f at a over shorter and shorter intervals next to a The same concept expressed in geometric term is the slope of the tangent line Rate Of Change Of f at a to the graph of f at the point a f 1 To express these results more concisely we use Leibniz A notation Let y f be a function of L The change in St between a variable point 5L and the xed point a is St 7 a Which is often denoted by Act The change m x Ax x 7 a The corresponding change in the function f is f 7 f a 7Which is often denoted by Ay The change m y Ay f 7 f a The average rate of change of f With respect to SC over the interval from a to SC is g f x 7 f a Ax x 7 a 39 The interval from a to SC shrinks to zero as an 7gt a equivalently A1 7 0 hence the rate of change of f at a can be expressed either by Ay Rate of change of f at a 7 A13101110 E or by Rate of change of f at a lim M 412 a 7 1 Here is another convenient way to express a rate of change at a Denote the change in J by b that is h Ax x 7 a Thenxah and f a h i f a h Rate of change of f at a lLin It is important to realize that the three expressions above for the rate of change of f at a are simply three alternative ways to express the same limit In a particular context one expression is likely to be more convenient or natural to use than another IMPORTANT Act is NOT a product of two numbers and Ag is NOT a product of two numbers The A notation is just a suggestive notation for keeping track of which variable is changing and by how much Derivative is simply a synonym for rate of change De nition The derivative of the function y f at a denoted by f a 7 is f a hm f x 7 f a 412 5570 provided the indicated limit exists We say f is differentiable at a if f 1 exists and is nite So far we have as sumed that instantaneous rates of change derivatives always exist Sometimes they do not Drop an idealized tennis ball What is its velocity at the instant it hits the ground and rebounds The derivative of f at a is just the rate of change of f at a As above we also have A h f w 312 A Z and f ltagt Continuity and Differentiability If f is differentiable at a then f is continuous at a

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