Reviewsheet2
.pdf
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Union High School**We aren't endorsed by this school
Course
MATH 1
Subject
Mathematics
Date
Jan 3, 2025
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3
Uploaded by fitapis902
Math 234 WESReview Sheet 2Fall 2014Topics1. Gradients2. Chain Rule3. Implicit Function Theorem4. Higher Order Partial Derivatives5. Clairaut’s Theorem6. Optimization(a) First and second derivative test(b) Constrained opitimizationi. Lagrange Multipliersii. Solve boundary for one variable and substitute into the original func-tion to get a function of one variableiii. Parametrize boundary and substitute into the original function to geta function of one variable7. Double integrals in Cartesian coordinates (i.e., no polar coordinates)(a) Over rectangles(b) Over regions that are not rectangles
Problems:1. Find the direction of maximal increase off(x, y) = 2x3y+ 5 ln(y)x2at thepoint (1,1). In what directions could you move so that the value offdoes notchange? Give your answer as a vector.2. Letf(x, y) =x2+y2. Where onx2+y2= 1 is#»∇fparallel toh2,-3i?3. Graph the zero set off(x, y) =x3-x2y. Explain, using the Implicit FunctionTheorem, precisely how you know from the zero set that the point (0,0) is acritical point.(i.e., it is not sufficient to simply say:“because the zero setintersects itself”)4. Consider the surfacef(x, y) =x2+y2. For which values ofxdo the level setsoffdefineyas an implicit function ofx? For these values, finddydx. For whichvalues ofydo the level sets offdefinexas an implicit function ofy? For thesevalues, finddxdy.5.(a) Ifg(x, y) =f(u, v, z), whereu= 3x+ 5,v= 2x+y2, andz=xy, find allfirst- and second-order partial derivatives ofg.(b) Ifg(x, y) =f(x2+ 2xy+y2), wheref(t) is a function of one variable, findall first- and second-order partial derivatives ofg.6. Find and classify all critical points off(x, y) =x3+ 2x2y2. Are any of theselocal extrema also global extrema? Why/why not?7. Consider the curveg(x, y) =x2-y3+y4= 0.Its graph is the loop in thefollowing figure.Letf(x, y) =x-y.(a) Without doing any computations, is there any guarantee thatf(x, y) ac-tually attains a global minimum value and a global maximum value some-where on the constraintg(x, y) = 0? Explain!(b) In the above figure, draw an assortment of level sets forf(x, y). (Labelthe levels.)(c) Mark theapproximatelocation wheref(x, y) attains its minimum valueong(x, y) = 0.(d) Mark theexactlocation wheref(x, y) attains its maximum value ong(x, y) = 0.
-0.500.51-0.50.511.5(e) Set up (but don’t solve!) the Lagrange system(~∇f(x, y) =λ~∇g(x, y),g(x, y) = 0.(f) Is (0,0) a solution of the Lagrange system from(e)?(g) What is~∇g(0,0)?(h) What is the moral of this problem?8. Find the global extrema off(x, y) =x2+3xy+y2on the rectangle{(x, y)|-2≤x≤2,0≤y≤3}.9. FindRRR(4-y2) dAwhereR={(x, y)|0≤x≤3,0≤y≤2}.10. Find the volume underf(x, y) = 1 over the regionDbounded byy=xandx=y2.11. Find the volume underf(x, y) = 6xover the regionDbounded byx2+y2= 4,x≥0.12. Reverse the limits of integration (you do not need to evaluate):R2√ln 30R√ln 3y/2ex2dxdy.