Finalguide - Summary Engineering Calculus III PDF

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Mathematics 2153-002 Calculus III Fall 2013 T, Th 9:30 AM - 10:45 PM, Rawl 102 Final Guide (1) Let x = t 3 + 6t , y = t 2 + 2t − 1. Determine the values of t for which this curve concaves up. (2) Let x = r (θ − sin θ), y = r (1 − sin θ). Find d y /d x for all values of t where dx /d t is not zero. Find the equation of the tangent line to the curve at the point (x(π/6), y(π/6)). (3) Compute the area under the curve given by x = t 3 , y = t 4 +5 between the values x(1) and x(2). (4) Compute the length of the curve given by the following equations: x(t ) = 3t 2 + 1, y = 5t 2 − 15,1 ≤ t ≤ 2. (5) Compute the are of the surface obtained by rotating around the x-axis the curve given by the following equations: x(t ) = 3t 2 + 1, y = 5t 2 + 15,1 ≤ t ≤ 2. (6) Compute the area bounded by the curve r = θ for 0 ≤ θ ≤ π/2. (7) Compute the arc length of the curve r = θ for 0 ≤ θ ≤ π/2. (8) What is an equation of the line passing through points (1,2,3) and (4,5,6)? (9) Compute the arc length of a curve (t 2 + 1, t 2 − 1,5t ) from t = 0 to t = 5. (10) Parametrize the curve (t 2 + 1, t 2 − 1,5t ) with respect to arc length. (11) Compute the curvature of the curve (t 2 + 1, t 2 − 1,5t ). (12) Compute the gradient of f ( x, y, z) = e x y z ( x yz) at (1,2,3). (13) Compute the derivative of f ( x, y, z) = e x y z ( x yz) in the direction (4,5,6) at (1,2,3). (14) Critical points: #1, 3, 7, page 892. (15) Absolute maxima and minima for a continuous function on a closed and bounded region: #31, RR 33, page 893 (16) Compute R x 2 + y 2 − x yd A, where R = {0 ≤ x ≤ 1,0 ≤ y ≤ 1}. (17) Double integrals over more complicated regions and volumes of solids: #25-#28, page 922. (18) Using double integrals to compute the area and the center of mass: #1, 3, 7, page 931. (19) Double integrals using polar coordinates: #1, 7, page 939. (20) Triple integrals in rectangular coordinates: #5, page 954. (21) Cylindrical coordinates: #1, 5, page 962. (22) Triple integrals using cylindrical coordinates: #25, 27, page 962. (23) Spherical coordinates: #1, 3, page 969. R RR (24) Triple integrals using cylindrical and spherical coordinates: compute Q x y zdV , where Q is the region between the spheres of radii 1 and 2 centered at the origin.

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