# MT1 pra - Math 53 Review for the ﬁrst midterm Sections...

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Unformatted text preview: Math 53, Review for the ﬁrst midterm Sections 210, 213; GSI: Ivan Mati´ c 1. Let z = 6 − x − y and 2x − y − z = 0 be the equations of two planes. (a) Find the equation of the line in which the two planes intersect. (b) Find the angle between the two planes. 2. Draw the curve v (t) = (cos t, sin2 t, sin t) and ﬁnd the tangent line to the curve at t = 0. 3. Find the equation for the set of points p which are equidistant from the plane z = 1 and the point (0, 0, −1). Sketch and describe the surface consisting of the points p. 4. Sketch the curve given by r = 1 − sin(2θ) and ﬁnd the area of one loop of the curve. 5. Find the area enclosed by the curve given parametrically by x = 1 2 sin 2t, y = sin t, 0 ≤ t ≤ π . 6. Find the area of the region in the plane bounded by the curve r = θ, 0 ≤ θ ≤ 2π and the positive x-axis. 7. Calculate the limit if it exists: lim x2 y 2 x6 + y 6 . (x,y )→(0,0) 8. Evaluate the limits, or show that they don’t exist: (a) (b) (c) (d) (e) (f) (x,y )→(0,0) lim lim xy ; x2 + y 2 xy 2 ; (x,y )→(0,0) x2 + y 4 (x,y )→(0,0) x2 lim xy 3 ; + y4 |x| + |y | + |z | · (xy + yz + zx); x2 + y 2 + z 2 (x,y,z )→(0,0,0) lim x sin3 y ; (x,y )→(0,0) x2 + y 4 lim x3 (y 2 + z 2 ) . (x,y,z )→(0,0,0) x4 + y 4 + z 4 lim 9. Assume that α is a real number between 0 and 1. (a) If x, y > 0, prove the inequality αx + (1 − α)y ≥ xα y 1−α . (Hints: You can either set αx + (1 − α)y = c and use the Lagrange multiplyers to ﬁnd the maximum of xα y 1−α or you can apply ln to the both sides of the inequality and use the fact that ln x is concave.) (b) What inequality do you get if you set α = 1/2? If you in addition put x = a2 and y = b2 , what is the name of the obtained inequality? (c) What inequality do you get if you set α = 3/4, x = a4 and y = b8 ? (d) Calculate x3 sin3 y . (x,y )→(0,0) x4 + y 8 lim ...
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