Exam. There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work! Best 5

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1 Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus III June, 06 Name: Exam There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work! Best 5

2 . Consider the curve given by the parametric equations x = t, y = t t. a. Which of the following points is on this curve 0,0, 7,4, 7,? A point a,b is on the curve given by the parametric equations if there exists t such that a = t, b = t t. If 0,0 is on the curve, then 0 = t implies t = ±. Since 0 ± ±, the point 0,0 is not on the curve. If 7,4 is on the curve, then 7 = t implies t = ±. Since 4 = and 4, we have for t = The point 7,4 is on the curve. 7 = t, 4 = t t. If 7, is on the curve, then 7 = t implies t = ±. Since ± ±, the point 7, is not on the curve. b. Find an equation of the line tangent of the curve at the point determined in a. A line given by y = cx+m is the line tangent at the point 7,4, if c is the slope of this line at 7 }{{} x, 4 }{{} y, i.e. c = y x and the point 7,4 is on this line, i.e. 4 = c 7+m. Since x t = t and y t = t, we have that the slope of the tangent line at 7,4 is c = y x = = 6 6 = The equation 4 = 9 7+m implies m = 4 = 9, thus the line tangent at 7,4 is y = 9 x

3 . Consider the points P =,,, Q =,,5, R =,6,. State the equation of the plane P through the points P, Q, R. A normal vector of the plane containing the points P, Q, R is the cross product of the vectors PQ =,, 5 = 0,0, andpr =, 6, = 0, 4,0 : 0,0, 0, 4,0 = 0 4 0,0, ,, ,0, =,0,0 An equation of the plane passing through the point P =,, and orthogonal to the vector,0,0 is x +0y +0z 0 = 0 = x =. Determine the point which lies not on the plane P: P =,,, Q =,,, R =,0,0. A point a,b,c is on the plane x + 0y + 0z 0 = 0, if a + 0b +0c 0 = 0, i.e. a,b,c is on the plane if a =. P =,, is on the plane, because the first component of this point is. Q =,, is not on the plane, because the first component of this point is not. R =,0,0 is on the plane, because the first component of this point is. Find an equation of the plane T passing through the point determined in and parallel to the plane P. Since the plane given by x = is orthogonal to the vector,0,0, we have that each plane parallel to the plane x = is orthogonal to,0,0. An equation of the plane T passing through the point Q =,, and orthogonal to the vector,0,0 is x +0y +0z = 0 = x =. 4 Exercise for Extra Credit points Find the distance between the planes T and P. The equations x =, and x = represent the planes parallel to the xz-plane and resp. units from it. Thus the distance between the planes is.

4 . a. Which geometric object is defined as the set of all points x,y in a plane such that 4x 9y = 6? Specify the location of the foci, and sketch a graph of this object. Describe the relationship between the foci and a point x,y which satisfies this equation. 4x 9y = 6 = 4 6 x y = = x 9 + y 4 = = x + y = 4x 9y = 6 represents an ellipse x a + y b =, where a =, and b = with the foci at c,0 and c,0 where c = a b = 9 4 = 5 5,0 0, x,y 5, ,- Let x,y be a point of the ellipse given by x + y =. If d is the distance from the focus at 5,0 to the point at x,y and d is the distance from the focus at 5,0 to x,y, then d +d = a = = 6. b. Which geometric object is defined as the set of all points x,y in a plane such that 4x +9y = 6? Specify the location of the foci, and sketch a graph of this object.

5 4x +9y = 6 = 4 6 x y = = x 9 y 4 = = x y = 4x 9y = 6 represents a hyperbola x a y b =, where a =, and b = with the foci at c,0 and c,0 where c = a +b = 9+4 =,0, c. Exercise for Extra Credit points Which geometric object is defined as the set of all points x,y in a plane with the following property: If l is the distance form x,y and 5,0, and l is the distance form x,y and 5,0, then l l = 8 or l l = 8 Write an equation, and sketch a graph of this object. Bythedefinitionahyperbolaisthesetofallpointsx,yinaplanesuchthatthedifference of the distances from two fixed points foci is constant. If 5,0 and 5,0 are the foci ±c,0 = ±5,0 l is the distance form x,y and 5,0, and l is the distance form x,y and 5,0, such that l l = a = 8 i.e. a = 4 then the equation of the hyperbola is x 4 + y b =, where c = a +b, i.e. 5 = 4 +b implies b = 5 6 = 9 =. Thus an equation of the hyperbola is x 4 + y =.

6 4. Consider the following vectors in R Find the cross product v w. v = 0,, and w =,0,. v w = 0,0,0 0 0,, ,0, =, 4, All vectors in R orthogonal to v and w are scalar multiple of the vector v w. Explain why this is true. If a,b,c 0 is orthogonal to v and w, then a,b,c is a normal vector of the plane P containing the vectors v and w. Since v w =, 4, is also a normal vector of the plane P the vectors, 4, and a,b,c are parallel. Thus a,b,c is a scalar multiple of, 4,. Find a vector u perpendicular to v and w, such that the volume of the box determined by v, w, and u is 58. Since u is perpendicular to v and w, the volume of the box determined by v, w, and u is the volume of the box determined by v, w, and u = the area of the parallelogram spanned by v, w, The area of the parallelogram spanned by v and w is v w =, 4, = = 9. the length of u From we know that u is a scalar multiple of v w, thus there exist c such that u = c, 4, = c, 4c,c. Since the length of u is u = c + 4c +c = c 9 = c 9 we obtain that if c = 58 9 the volume of the box determined by v, w, and u =, so c = ±. For the vector u we have = 9 c 9 = c 9 = 58 u =, 4, = 6, 8,4, or u =, 4, = 6,8, 4.

7 5. An object moves on the helix cost, sint, t, for t 0. a Let f be a differentiable function with ft 0. Explain why the position function cosft, sinft, ft describes motion along the helix. The position function rt = cos ft, sin ft, ft describes motion along the helix, if each point of the curve of rt, i.e. the point of the form cosft, sinft, ft is on the helix given by cost, sint, t The curve of the position function cost, sint, t, for t 0 contains the points of the form cos t, sin t, t, where t is a non-negative real number. By the assumption the real number t = ft is non-negative for each t. The point cosft, sinft, ft = cost, sint, t is of the form described above for all t 0. Therefore each point of the curve of the position function cosft, sinft, ft is on the helix given by cost, sint, t. b Find a position function r that describes the motion along the helix if it occurs with the speed. Find the length of the curve for 0 t, and for t. A position function that describes the motion along the helix is of the form rt = cosft, sinft, ft Themagnitudeofthevectorr t = cosft, sinft, ft = sinft f t, cosft f t, f t gives the speed of the object at the time t. We have r t = sinft f t, cosft f t, f t = sinft f t +cosft f t +f t = f t sin ft+cos ft +f }{{} t = f t +f t = f t = f t = if f t =, thus f t =, and consequently ft = function rt = cos t, sin t, dt = t. The position t

8 describes the motion along the helix where it occurs with the speed. b b b Since the length of the curve for a t b is r t dt = dt = t = a b we a a a have the length the length of the curve = dt = t =, and of the curve = dt = t =. for 0 t 0 0 for t c Find a position function r that describes the motion along the helix if it occurs with the speed t. Find the length of the curve for 0 t, and for t. A position function that describes the motion along the helix is of the form rt = cosft, sinft, ft The magnitude of the vector r t = sinft f t, cosft f t, f t gives the speed of the object at the time t. We have r t = sinft f t +cosft f t +f t = f t sin ft+cos ft+f t = f t +f t = f t = f t = t if f t = t, thus f t = t, and consequently ft = t = t. The position function t t rt = cos, sin, t t dt = t dt = describes the motion along the helix where it occurs with the speed t. The length of the curve for a t b is have the length of the curve for 0 t = 0 b a =, and r t dt = b a the length of the curve for t t dt = t b a = d Exercise for Extra Credit points Interpret the results from b and c. The position functions = b a we = 7

9 cos t, sin t, t, and t t cos, sin, t describe the motion along the same helix with speeds, and t respectively. The particle moving along the helix given by cos t, sin t, t is at time 0,, at the positions,0,0 = cos cos, sin 6 cos 6, sin 0, sin,, 6 0, 0 Since the speed is constant, the distance along the helix between two points of the same time interval is the same. The time intervals from t = 0 to t =, and from t = to t = is the same, namely 0 = =, therefore the distance along the helix from,0,0 to cos, sin, as well as from cos, sin, to cos 6, sin 6, 6 is the same, namely.

Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space.

10 VECTOR FUNCTIONS VECTOR FUNCTIONS Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space. Here, we prepare the way by developing