Parametrics

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School
University of California, Berkeley**We aren't endorsed by this school
Course
MATH 53
Subject
Mathematics
Date
Dec 17, 2024
Pages
2
Uploaded by ElderRoseSeahorse30
Math 53 - 20242CALCULUS WITH PARAMETRIC CURVESParametrics!1Parametric Equations1.(10.1.26)Use the graphs ofx=f(t) andy=g(t) to sketch the parametric curvex=f(t), y=g(t).Indicate with arrows the direction in which the curve is traced astincreases.2.(10.1.9)Sketch the curve by using the parametric equations to get asymptotic data: what happenswhentis large (what is the relationship betweenxandy? - what part of the equation can you throwaway?) what about whentis small? Plot points to determine the behavior neart= 1.x=t, y= 1t.3.(10.1.13)Do the same as above, then eliminate the parameter to verify your work!x= sint, y= csct,0< t < π/2.4. See if you can write out parametric equations qualitatively for the following curves:(Hint: For the first think about what the curve looks like from far away. For the second think aboutthe ranges ofxandyand what standard functions have those ranges)2Calculus with Parametric Curves5. Letx(t), y(t) represent where you are standing at timet. Explain, using this analogy whydy/dxonyour path at a certain time is equal tody/dtdx/dt.What is your direction vector at a certain timet?
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Math 53 - 20244ACKNOWLEDGEMENTS6.(10.2.7)Find an equation of the tangent line to the parametric curvex= 1 + lnt, y=t2+ 2,(t >0)at the point (1,3) by two methods: find the derivative first, then eliminating the parameter and b) byfirst eliminating the parameter.7.(10.2.14)Finddy/dxandd2y/dx2forx=t2+ 1, y=et1.For which values oftis the curve concave upward? Make sure you know what the second derivativemeans!!8. Explain geometrically why the arc length formula is the way it is. (Hint: think about the directionanalogy from above and DRAW A PICTURE!)3Vectors9. Draw the curvex(t) = cost,y(t) = sint. On the same graph, draw the vector (an arrow) from theorigin to the point (x(π/4), y(π/4)). Think about how this arrow changes as you changet.10. Letr(t) = (x(t), y(t)) from above. On the same graph as above for some smallϵ >0 draw the vectorsr(π/4±ϵ)and draw their difference. (The sum of two vectors is the separate sums of the components and alsothe arrow you get by first following one and then the other tail to tip.) What does this result feel like?11. What angle does the position vectorr(t) make with thetangentvectorr(t)? Which property of thecurve is making that happen?12. (Capstan Problem) You and a friend are going rock climbing. All is going well until you go to holdthe line for your friend who you are now realizing weighs significantly more than you. You worry thatif they fall you will be lifted into the air instead of holding them up. However, your friend assures youthat since the rope is looped around a cylinder with non-negligible friction you will be able to lowerthem down safely. As it turns out they were right (luckily for everyone!) Use the fact that the forceof friction isµtimes the force orthogonal to the wire (and points in the direction of the wire) andthat tension pulls on each bit of wire (in the direction of the wire) to find out just how much forcemultiplication a wire loopedθradians around a cylinder provides.What changes if instead of a cylinder we used a parametric curver(s) parametrized by arc length?(That is the arc length between 0 andsalongrissfor alls.)4AcknowledgementsAll problems labeled with bold numbers are based on Stewart, James,Multivariable Calculus, CengageLearning, 8th ed.
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