Calculus III - Problem Drill 08: Clindrical and Quadric Surfaces Question No. of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the. Which one of the following equations in gives a clindrical surface as its graph? (A) + + = 0 Question #0 (B) 5 = 0 (C) + 4 = 0 (D) = e A. Incorrect! This equation has an empt set as its graph. B. Incorrect! Presence of all three variables in the equation makes it not clindrical. C. Incorrect! Presence of all three variables in the equation makes it not clindrical. D. Correct! This is a clinder. The ais of clinder is -ais. Its generator is a curve other plane = constant. = e, on - plane, or on an E. Incorrect! e of (A) ~ (D) is correct. The variable is missing from the equation, so the surface etends indefinitel in -direction. The rulings are straight lines parallel to -ais, and the all pass through the generator = e. The computer generated graph is displaed below.
Question No. of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the. Which one of the following equations in gives a hperboloid of one sheet as its graph? Question #0 (A) + 5 4 = (B) + = 4 8 (C) = 0 5 (D) + = A. Correct! It is in the form + = with b c a 6 a =, b= and c = 5 5 B. Incorrect! This equation is in the form: + + = C. Incorrect! This equation is in the form: + = D. Incorrect! This equation is in the form: c + = b E. Incorrect! e of (A) ~ (D) is correct. + 5 4 = + = So ( ) ( 5) ( 4) 6 a =, b= and c = 5 5 Trace-wise: = 0 + =, an ellipse. ( ) ( 5) = 0 =, a hperbola. ( 5) ( 4) = 0 =, a hperbola. The graph is shown below. ( ) ( 4)
Question No. of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the. Which one of the following equations in gives an elliptic cone as its graph? (A) + = 4 (B) + = 8 + 6 9 Question #0 (C) + = 9 5 (D) = 7 6 5 A. Incorrect! This equation is in the form: B. Incorrect! This equation is in the form: a + = c + + = C. Incorrect! This equation is missing one of the variables. D. Incorrect! This equation gives an empt set as its graph. E. Correct! None of (A) ~ (D) is correct. A. B. C. + = + = is an equation of an elliptic paraboloid. 4 + = 8 + 6 + + = is an equation of an ellipsoid. 9 4 ( ) ( ) ( ) + = 9 + = is an equation of an elliptic clinder. 5 5 ( ) D. = 7 is impossible to meet in 6 5, since 0 (,, ) 6 5
Question No. 4 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 4. Which one of the following equations in (A) + 4 = 0 (B) + 5 = 0 gives a hperbolic paraboloid as its graph? Question #04 (C) = 4 (D) + + 5 = 0 A. Incorrect! This equation is in the form: B. Incorrect! This equation is in the form: C. Correct! This equation is in the form: a + = b + = = c a D. Incorrect! This equation gives the origin as its graph. E. Incorrect! e of (A) ~ (D) is correct. = 4 = 0 =, a parabola. = 0 =, a parabola. 4 = =±, a pair of lines. = 0 > 0 4 =, a hperbola. 0 0 0 = 0 < 0 4 = ( ) ( ) 0 0, a hperbola. We get:
Question No. 5 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the Question #05 5. Which one of the following equations in (A) + = (B) + + = (C) + + = (D) + = 0 gives a hperboloid of two sheets as its graph? A. Correct! This equation is in the form: B. Incorrect! This equation is in the form: C. Incorrect! This equation is in the form: D. Incorrect! This equation is in the form: E. Incorrect! e of (A) ~ (D) is correct. + = b c a + = b c a + + = + = b c a + = = ( ) ( ), a hperbola. = 0 = ( ) ( ) = 0 =, a hperbola. ( ), an ellipse. = 0 > + = > 0 ( ) ( ) = 0, ±, 0 = 0 < no graph. on a graph.
Question No. 6 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 6. Which one of the following equations in gives an ellipsoid as its graph? Question #06 (A) + + = (B) + 5 = 4 (C) + + = 0 9 7 (D) 4+ + + = A. Incorrect! This equation is obtained b translation along ais of one of the quadric surface. Which one is it? And how much shifting is taking place? B. Incorrect! Two pluses and one minus for coefficients of squared terms make it no ellipsoid. C. Correct! This gives an origin. D. Correct! This is trick. Since the problem does not ask to identif canonical form, this is indeed an ellipsoid, with translations. E. Incorrect! e of (A) ~ (D) is correct. B completing the squares for and, we get: ( ) ( + ) 4+ + + = + + = + 4+ = 7 We see that this equation represents an ellipsoid in canonical form, if we shift the and aes b to positive -direction and down to the negative -direction. i.e., with and +, we get: + + = 7 + + =, which is a canonical ellipse in the translated coordinate 7 4 sstem. ( ) ( + ) So + + = or 7 4 ( 0,, ) in the original coordinate sstem. 4+ + + = is an ellipsoid, centered at
Question No. 7 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 7. Which one of the following equations in gives an elliptic paraboloid as its graph? Question #07 (A) + = 9 7 (B) = 9 7 (C) + = 8 4 6 (D) + = 4 4 6 A. Incorrect! You shall get a hperbolic paraboloid with shifting. B. Correct! This involves shifting along -ais. C. Incorrect! This shall give ou a two sheet hperboloid, shifted 4 units upward in positive -direction, of the canonical form. D. Incorrect! This shall give ou a one sheet hperboloid. E. Correct! None of (A) ~ (D) is correct. = = + 9 7 9 7, then we obtain the canonical form of an elliptic paraboloid in translated So if we let coordinate sstem. = + 9 7 Sketch = + and shift the graph units toward positive -direction. 9 7
Question No. 8 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 8. Which one of the following surfaces is the one for the equation (A) (B) + =? 4 5 4 4 Y Question #08 (C) (D) Y Y Y A. Incorrect! You need to check the sign of term. B. Correct! Shifting of the canonical form in the negative -direction is taking the place for this equation. C. Incorrect! This one is the canonical version of the equation. D. Incorrect! You need to check the term involving. E. Incorrect! e of (A) ~ (D) is correct. + = + = + = 4 5 4 4 4 4 5 4 5 4 So b letting +, we see that paraboloid in the shifted coordinate sstem. This means that the graph of ( 54) one unit to the negative -direction. Which gives: ( ) = is in canonical form of the equation of hperbolic + = is obtained from that of 4 5 4 4 = b shifting it ( 54)
Question No. 9 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 9. Which one of the following is the level curves at several heights of the surface described b =? 4 (A) (B) Question #09 (C) (D) A. Correct! The level curves are hperboloids, including the degenerate one. B. Incorrect! The equation does not generate ellipses for level curves. C. Incorrect! The equation does not generate parabolas when is set constant. In fact, these are for the positive branch of =, i.e., for 4 = 4+. D. Incorrect! The equation does not generate straight lines when is set constant, ecept for = 0, but there will be a pair of lines through origin. E. Incorrect! e of (A) ~ (D) is correct. Projection onto - plane of traces of the surface on the plane = 0 for various constants 0 is b definition the level curves for the surface at the height 0. This surface is for a hperbolic paraboloid. Y Level Curves at =-0, -5, 0, 5 and 0
Question No. 0 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 0. Which one of the following equations describes an elliptic cone that is obtained from translation of the canonical form? Question #0 (A) (B) (C) = + + = + = + (D) = + + A. Incorrect! This, once again is a two sheets hperboloid, with a translation. B. Incorrect! This gives a shifted hperbolic paraboloid. C. Incorrect! This is a one sheet hperboloid. D. Incorrect! This gives a shifted elliptic paraboloid. E. Correct! None of the given answers is correct. The elliptic cone has the standard form: + = As for = + +, we have ( ) So ou get ( ) = + = + + + = and this is a two sheets hperboloid with a translation. As for translation. = +, we have = and this is a hperbolic paraboloid with a As for form. As for = +, we have = + +, we have + =, and it is a one sheet hperboloid in canonical ( ) ( ) + = + = + = +, which is a an elliptic paraboloid with a translation. If, for instance, we consider = + +, then ( ) + = + = +, and hence this would have been an elliptic cone with a translation.