Chapte Sample Exam Poblems maked with an asteisk (*) ae paticulaly challenging and should be given caeful consideation.. Conside the paametic cuve x (t) =e t, y (t) =e t, t (a) Compute the length of the cuve fom t =to t =. (b) Compute the length of the cuve fom t =to t =. (c) Convet this paametic equation into an equation of the fom y = f (x), and compute the length of the gaph of f (x) fom x =to x =. (d) Is the answe in pat (c) the same as the answe in pat (b)? How do you explain this esult?. Conside the paametic cuve x (t) =a cos t, y (t) =b sin t, t fo a, b. (a) Sketch and identify this cuve. (b) Set up an integal to compute the length of this cuve.. The cuve below can be paametized by x (t) =t sin t, y (t) =t +cost. y _5 5 5 5 x (a) Give a paametization fo the cuve below. (b) Give a paametization fo the cuve below. y y _5 5 5 5 x _5 5 5 5 x 67
CHAPTER PARAMETRIC EQUATIONS AND POLAR COORDINATES 4. Match each ectangula gaph of = f (θ) in the left column below with the coesponding pola gaph in the ight column. Explain you easoning. (a) (i) _ 4 5 6 7 O _ (b) (ii) _ O _ 4 5 6 7 (c) (iii) _ 4 5 6 7 _ O (d) (iv) _ 4 5 6 7 O _ 68
CHAPTER SAMPLE EXAM 5. Conside the pola cuve = cos θ sin θ, θ< 4. (a) What happens to as θ appoaches 4? (b) Convet the above equation to an equation y = f (x) in ectangula coodinates. (c) What value of θ will give the point (, )? (c) What is the domain of this pola cuve when viewed as a function f (x)? 6. Conside the pola cuve = f (θ) shown below. Note that f (θ) 5 fo all θ and that f () = f (). 4 =f( ) _4 _ O 4 4 (a) Show that the aea enclosed by the gaph of f (θ) must be geate than. (b) Must the aea enclosed by this cuve fom θ be finite? Why o why not? (c) Find a function g (θ) satisfying the same conditions as f (θ) above, such that the aea enclosed by g (θ) is 4. 7. Conside the cuve x = e t 5 cos t, y = e t 5 sin t, t. (a) Find the values of t whee the line tangent to the cuve is vetical. (b) Find the values of t whee the slope of the line tangent to the cuve is. 8. Find the aea unde the paametic cuve x =sint, y =costsin t, t. e 9. Conside the pola equation = +ecos θ. (a) What is the diectix fo this pola cuve? (b) Classify the pola cuve if (i) e =. (ii) e = (iii) e =.9 (iv) e =. (c) Classify the pola cuve =, and compute the eccenticity. 4+cosθ. Conside the paametic cuve x =t 5 +, y =t, t R. (a) Find the values of t whee the slope of the line tangent to the cuve is. (b) What value of t gives the point (, ) on this cuve? (c) Is dy/dx defined at (, )? Descibe the shape of the cuve at this point. 69
. Conside the family of conics x k + y k =, k>, k. (a) Descibe the conic if k>, and compute the foci. (b) Descibe the conic if <k<, and compute the foci. (c) Find the value of k andanequationfotheconicinthisfamilyifthey-intecepts ae 5 ±.. Conside the paametic cuve x = t cos t, y = t sin t, t >. (a) Wite an integal which descibes the length of this cuve fom t =to t =. (b) What substitutions could be used to evaluate the integal in pat (a)? Do not evaluate the integal. (c) Set up, but do not evaluate, an integal which descibes the suface aea obtained by otating this cuve about the x-axis. Chapte Sample Exam Solutions. (a) L = x (t) + y (t) dt = (b) L = e t +4e t dt = 5 ( e e ) (c) y =x, a staight line with slope. L = e t +4e t dt = 5 et dt = 5(e ) ( ) +(4 ) = 5 (d) They ae diffeent, since in pat (b), we ae computing the distance fom (e, e) to ( e, e ) and in pat (c) fom (, ) to (, 4).. (a) This is the uppe half of the ellipse x a + y b =. (b) L = x (t) + y (t) dt = a sin t + b cos tdt. (a) x (t) = x (t) = (t sin t), y (t) =y (t) =t +cost (b) x (t) = x (t) = t sin t, y (t) = y (t) = (t +cost) 4. The matches ae (a) (iii), (b) (ii), (c) (iv), and (d) (i). 5. (a) lim = lim θ (/4) θ (/4) cos θ sin θ = (b) We have cos θ sin θ =.Thisgivesx y =o y = x. (c) θ =gives the point (, ). (d) The domain is x. 6. (a) The aea enclosed by f (θ) is geate than the aea of a cicle of adius. Thisaeais4.6 >. (b) The aea enclosed by f (θ) is less than the aea of a cicle of adius 5. This aea is 5 78.5 < 8. Sotheaeaisalwaysfinite. (c) We want f (θ) dθ =4. Iff (θ) is a constant C, weneed C =4 C =4 C = 4. So choose f (θ) = 4 fo all θ. 6
CHAPTER SAMPLE EXAM SOLUTIONS 7. (a) The tangent is vetical if dx dt = et 5 cos t e t 5 sin t =. Thus, cos t sin t = tan t = t = 4 o 5 4. (b) dy dx = dy/dt dx/dt = et 5 (sin t +cost) e t 5 = sin t +cost =sint cos t cos t = (cos t sin t) t = o. 8. A = y (t) x (t) dt = cos t sin t cos tdt = cos t sin tdt= cos t = ( ) + = 9. (a) x =is the diectix fo this conic. (b) (i) Hypebola, e> (ii) Paabola, e = (iii) Ellipse, <e< (iv) Ellipse, <e<, but nealy cicula. (c) = 4+cosθ = + 4 cos θ. This is an ellipse with eccenticity e = 4.. (a) Set dy dx = dy/dt dx/dt = t 5t 4 = t =. Hence, at t = ±, the slope is. (b) x (t) =, y (t) = gives t 5 ==t,ot =. (c) At t =, dy dx = is not defined. In this case, the line tangent to the cuve is vetical. t. (a) If k>, k > and the conic is an ellipse with a = k, b = k. Hencea>b,and c = a b = =. So the foci ae at (, ) and (, ). (b) If <k<, k is negative, k > and x k y k =is a hypebola. c = a + b = k + k = =. So the foci ae at (, ) and (, ). ( (c) The points, ± ) 5 y 5 ae on this ellipse. So k = 4 k =, 5 4 = k, k = 9 4, k =.The equation of the ellipse is x 9/4 + y 5/4 =.. (a) L = x (t) + y (t) dt = (cos t t sin t) + (sin t + t cos t) dt = +t dt,afte simplifying. (b) Substitute t =tanu, dt =sec uduto obtain L = tan sec udu (c) S = y (t) x (t) + y (t) dt = t sin t +t dt 6