{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math_20C_Review

Math_20C_Review - Math 20C Summer 2010 Review Example 1 The...

This preview shows pages 1–4. Sign up to view the full content.

(8/31/10) Math 20C. Summer 2010. Review. Example 1 The three forces, F 1 = < 2 , 1 , 2 >, F 2 = < 1 , 1 , 1 > , and F 3 = < - 2 , - 3 , - 1 > , measured in pounds, are applied at the same point on an object. What is the magnitude of the combined force? Answer: F 1 + F 2 + F 3 = < 1 , 1 , 2 > [Magnitude] = 1 2 + 1 2 + 2 2 = 6 pounds Example 2 Which angle in the triangle with vertices P = (1 , 1 , 1) , Q = (3 , 4 , 2), and R = ( - 1 , 2 , 2) is the right angle? Answer: The angle at P is the right angle because the dot product of −→ PQ = < 2 , 3 , 1 > and −→ PR = < 2 , 1 , 1 > is zero. Example 3 Is the smallest positive angle θ between the vectors A = i - j and B = j + k an acute angle, a right angle, or an obtuse angle? Answer: The angle is obtuse, because cos θ = A · B | A || B | = 1 2 2 = 1 2 is negative. Example 4 Give parametric equations of the line that is perpendicular to the surface x 2 + y 3 + z 4 = 3 at (1 , 1 , 1). Answer: ( x 2 + y 3 + z 4 ) = < 2 x, 3 y 2 , 4 z 3 > equals < 2 , 3 , 4 > at (1 , 1 , 1). Line: x = 1 + 2 t y = 1 + 3 t z = 1 + 4 t Example 5 Give an equation of the plane that contains the lines, L 1 : x = 2 + t y = 3 + t z = 4 - t and L 2 : x = 2 + 6 t y = 3 z = 4 - 7 t. Answer: < 1 , 1 , 1 > × < 6 , 0 , 7 > = < 7 , 1 , 6 > Plane: 7( x 2) + ( y 3) 6( z 4) = 0 Example 6 What is the area of the triangle with vertices P = (0 , 0 , 0) , Q = (3 , 1 , - 2), and R = (1 , 4 , 0)? Answer: −→ PQ × −→ PR = < 3 , 1 , 2 > × < 1 , 4 , 0 > = < 8 , 2 , 11 > [Area] = 1 2 | −→ PQ × −→ PR | = 1 2 8 2 + 2 2 + 11 2 = 189 Example 7 The curve C : x = t 2 + t - 2 , y = cos(2 t )+2 , - 2 . 75 t 1 . 75 is shown in Figure 1 Draw the velocity vector at t = - 1, using the scales on the axes to measure its components. (Use the values cos( - 2) . = - 0 . 42 and sin( - 2) . = - 0 . 91.) x - 3 - 2 - 1 1 2 3 y 3 FIGURE 1 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Math 20C Review p. 2 Answer: R ( t ) = < t 2 + t 2 , cos(2 t ) + 2 > v = R ( t ) = < 2 t + 1 , 2 sin(2 t ) > R ( 1) = < 2 , cos( 2) + 2 > . = < 2 , 1 . 58 > v ( 1) = < 1 , 2 sin( 2) > . = < 1 , 1 . 82 > Figure A7 x - 3 - 2 - 1 1 2 3 y 3 Figure A7 Example 8 Give a definite integral that equals the length of the curve in Example 7. Do not simplify the integrand or attempt to carry out the integration. Answer: [Length] = integraldisplay 1 . 75 2 , 75 radicalbig (2 t + 1) 2 + ( 2 sin(2 t )) 2 dt Example 9 Figure 2 shows the graph of g ( x, y ) = 10 cos( xy ) 1 + 2 y 2 . Find, without using derivatives, the global maximum of z = g ( x, y ) and the values of ( x, y ) where it occurs. y x z FIGURE 2 Answer: [Global maximum] = 10 along the x -axis (where y = 0)
p. 3 Math 20C Review Example 10 (a) Draw the level curves of f ( x, y ) = radicalbig x 2 + y 2 where it has the values 1, 2, and 3. (b) Describe the graph of f . Answer: (a) Figure A10 (b) The graph is a circular cone with its vertex at the origin and the positive z -axis as axis. x 4 y 4 Figure A10 Example 11 Explain the shape of the graph of H ( x, y ) = 2 sin( y 2 ) + 8 1 + x 2 + y 2 in Figure 3. y x z FIGURE 3 Answer: z = 2 sin( y 2 ) is independent of x and oscillates increasingly rapidly between 1 and 1 as y moves away from 0, while z = 8 1 + x 2 + y 2 is a circularly symmetric bump with its highest point at x = 0 .y = 0.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}