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cal 3 practice test (not final (2)

# cal 3 practice test (not final (2) - Calculus III Practice...

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Calculus III Practice Test 3 F. Mokhtarian 1. Find a vector function that represents the curve of intersection of surfaces x 2 + y 2 = 4 and z = xy . 2. Let C be the curve given by ~ r ( t ) = ( t cos t, t sin t, t ). (a) Show that C lies on the surface of the cone z 2 = x 2 + y 2 . Use this to sketch the curve. Indicate the direction in which t increases on your graph. (b) Give parametric equations of the tangent line to the curve at P ( - π, 0 , π ). (c) Give the general equation of the normal plane to the curve at P ( - π, 0 , π ). (d) If a particle’s position vector is given by ~ r ( t ), what is particle’s speed at any time? 3. Let C be the curve given by ~ r ( t ) = (1 , t 2 , t 3 ). Find: (a) length of the curve for 0 t 1 (b) unit tangent and unit normal vectors (c) the curvature κ ( t ) (d) tangential and normal components of acceleration. 4. If ~u ( t ) = ~ r ( t ) · ( ~ r 0 ( t ) × ~ r 00 ( t )), show that ~u 0 ( t ) = ~ r ( t ) · ( ~ r 0 ( t ) × ~ r 000 ( t )) 5. Sketch and describe the following: (a) domain of f where f ( x, y, z ) = ln(9 - x 2 - 9 y 2 ) (b) level curve of g ( x, y ) = 1 x 2 - y 2 corresponding to c = 1 (c) level surface of f ( x, y, z ) = x 2 -

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cal 3 practice test (not final (2) - Calculus III Practice...

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