Lesson 2 MATH 203 - Week 2 Lesson MATH 203 Salik Bahar...

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Week 2 Lesson MATH 203Salik BaharJanuary 14, 2017CHAPTER 2.1 THE TANGENT AND VELOCITY PROBLEMS.Definition:Atangent lineof a curve isa line that touches a curvef(x) at one and only one pointPin some small intervalI.and has the same direction as the functionf(x) at the point of contact inI.11The equation of the tangent line of the formy=mx+bwheremis the slope of the tangent line andbis they-intercept.If we are given more than one point in the linear functiony=mx+b, then we can obtain1P1P2P3tangent line of f(x) at P1tangent line of f(x) at P2tangent line of f(x) at P3
the slopemby applying thepoint-slope formulaPoint-Slope Formulam=y2-y1x2-x1If are given the slope and a point ony=mx+b, then we can also find they-interceptbby replacingxandyfrom the equation with any coordinate point (x, y) then solve forb.With thepoint-slope formula, we can also rewrite the equation of the tangent line asm(x-x1) = (y-y1)Example:Find the equation of the tangent line to the curvef(x) =x2at the pointP(1,1).We can find the equation of the tangent liney=mx+bas soon as we know the slopem, but we are only given one pointP(1,1) on the tangent line.So we can’t apply thepoint-slope formula since we need at least two points on the tangent line.Thus we will compute an approximation ofmby choosing a nearby pointQ(x, x2) onf(x) =x2and computing the slope of the secant linePQ. Recall that asecant lineis aline that intersects a curve more than once.mPQ: ”slope ofPQmPQ=x2-1x-1wherex6= 1.The tables below displays the result ofmPQasQget closer and closer toPfrom theleft hand side and right hand side.xmPQxmPQ24011.52.50.51.51.12.10.91.91.012.010.991.991.0012.0010.9991.99911PQ
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Notice that the closerQis toP, then the closerxis to 1 and the closermPQis to 2.Therefore the slope of the tangent line,m2.We say theslope of the tangent lineis thelimit of the slopes of the secant linesand can use the notationlimQ!PmPQ=mlimx!1x2-1x-1= 2Sincem= 2 then the equation tangent liney= 2x+b. Now we can findy-intercept,b,by replacing (x, y) withP(1,1) into the equation.y= 2x+b)1 = 2(1) +b)b=-1Therefore, the equation of the tangent liney= 2x-1.THE VELOCITY PROBLEMSLetf(t) be a function oft.Definition: Average velocity=change in positiontime elapsed=Δf(t)Δt=f(t2)-f(t1)t2-t1whereΔis the ’the change in ...’.f(t)xyΔf(t)ΔtQP
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In other words, average velocity is the slope of the secant linePQ, wherePandQare onf(t).Note:average speed =distance travelledtime elapsedaverage velocity =displacementtime elaspedDefinition: The instantaneous velocityis the tangent line off(t) at some pointP(t, f(t))instantaneous velocity= limQ!PmPQPm=instantaneous velocityyx
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