Math 126C: Week 3 Review

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1 Math 126C: Week 3 Review Note: These are in no way meant to be comprehensive reviews; they re meant to highlight the main topics and formulas for the week. Doing homework and extra problems is always the best way to prepare and fully understand the material! Section 12.6 Cylinders: A cylinder is a surface that consists of all lines that are parallel to a given line and pass through a given plane curve. Typically (in our examples and homework problems), such surfaces have only two variables in the given equation. When this is the case, the cylinder looks like the graph of this equation in the two variables stretched out along the axis of the third variable that does not appear in the equation. Refer to Figures 1, 2, and 3 on p. 827 for a reference of what this looks like. Sketching quadric surfaces: The easiest way to sketch surfaces is to use traces; set one variable equal to a few di erent constant values, then plotting the resulting graphs in the other two variables. After this, try to stitch together the surface by putting these graphs into R 3. See Example 5 on p. 829 for a good visualization of this. Quadric surface classification: The table on p. 830 contains everything that you need to know. The axes of symmetry of the surfaces can be changed by switching variables, and the surfaces can be translated in the same way that spheres or lines are translated (i.e., changing x to (x 2) will shift the surface two units forward on the x-axis). Sections 10.1/13.1 Vector functions: A vector-valued function is a function whose domain is a set of real numbers and whose range is a set of vectors. Typically, these are of the form r(t) =hf(t),g(t)i in R 2,andr(t) =hf(t),g(t),h(t)i in R 3 with t ranging over some interval I. The set of all such points, C, is called a space curve. The components f(t),g(t),h(t) are called the parametric equations of C, and the variable t is called the parameter. We sometimes write these component functions as x(t),y(t),z(t) also. Eliminating the parameter t: In some cases with parametric equation in R 2, we can eliminate the parameter t by solving for t in terms of x or y and then plugging this expression into the other variable s expression in terms of t. This results in a normal graph of x and y, and the set of points on the graph is the same as the curve C. 1

2 Sections 10.2/13.2 Derivatives of vector functions: We can take derivative of vector functions r(t) by taking the derivative of each component function with respect to t: r 0 (t) =hx 0 (t),y 0 (t),z 0 (t)i Properties of derivatives of vector functions: See the table on p. 850 of the book. The most important facts are that derivatives of vector functions obey the product rule with respect to both the dot product and the cross product. Tangent lines: The tangent line to a curve r(t) at a point r(t 0 ) is the line passing through the point r(t 0 ) and parallel to the vector r 0 (t 0 ). This tangent line has the vector equation l(t) =r(t 0 )+tr 0 (t 0 ). In the case of r(t) being a two-dimensional vector, we can solve for the slope dy dx to the curve at a point (x, y) by using the chain rule: of the tangent line dy dx = dy/dt dx/dt Note that the tangent line is horizontal if dy/dt =0at the point and the tangent line is vertical if dx/dt =0at the point. Integrals of vector functions: Just as we can di erentiate vector functions, we can also integrate them. This is once again done component-wise: Z b a Z b r(t) dt = f(t) dt, a Z b a g(t) dt, Z b a h(t) dt Section 13.3 The arc length formula: If a curve has the vector equation r(t), then the length of the curve from r(a) to r(b) is given by the integral s Z b dx dy dz L = +. dt dt dt This can be written much more succinctly as a L = Z b a r 0 (t) dt 2

3 Math 126 Autumn (8 points) Identify the surface in R 3 given by the equation. You do not need to show any work. Choose your answer from the following: (a) 48x + y 2 =4z 2 cone ellipsoid elliptic cylinder elliptic paraboloid hyperbolic cylinder hyperbolic paraboloid hyperboloid of one sheet hyperboloid of two sheets parabolic cylinder (b) 4y 2 z 2 =1 (c) x 2 +6x y 2 4y + z 2 = 5 (d) x 2 +6x y 2 4y + z 2 =3

4 Math 126 Autumn (10 points) The graph below shows the curve ~r (t) =ht 2,kt t 3 i for t 0, where k is a constant. The line through the points A and B is tangent to ~r (t) ata. Thex-coordinate of A is 4 and B is the point (28, 0). Find the value of k. A r (t) B

5 2. Consider the shape determined by the equation x 2 + y 2 = z 2. (a) Draw an approximate sketch of the shape. (b) Show that the parametric path ((1 t) 2 cos(t), (1 t) 2 sin(t), (1 t) 2 ) lies on the shape. 5

6 (c) For which time(s) t is the speed of a particle moving along the path of part (b) minimized? (Recall that the speed is the magnitude of the velocity vector.) 6

7 4. (a) (10 pts) Consider the vector function r(t) =ht 2 2t, t 3 4ti and the corresponding parametric curve x = t 2 2t, y = t 3 4t. i. Find the value of d2 y at t = 1. dx2 ii. Find the value(s) of t at which the tangent line is parallel to the vector h1, 2i. (b) (5 pts) Find parametric equations for the tangent line to the curve given by r(t) =h2sin(3t), 3t, 2t cos(t)i at the time t = 3. (Give exact, simplified, numbers in your answer).

8 Math 126G First Midterm Spring (7 points) Let r(t) = 3 2t 1 i+ 1+t2 1+t j. Calculatetheintegral r(t) dt. Giveyouranswer 2 0 in exact form. 2 (8 points) Consider the curve in R 2 with parametric equations x =4t 2 +t+1, y = t 4 +2t. Give the coordinates of the points on the curve where the tangent line has slope 2.

9 Math 126 Spring (8 points) Let X be the surface in R 3 determined by the equation z 2 = x 2 2y 2. You are not required to show any work for the following questions. (a) Identify the traces of X in the indicated plane. i. The trace in the plane x =0isa(n): circle hyperbola point ellipse parabola pair of lines ii. The trace in the plane x = k (k 6= 0)isa(n): circle hyperbola point ellipse parabola pair of lines iii. The trace in the plane y =0isa(n): circle hyperbola point ellipse parabola pair of lines iv. The trace in the plane y = k (k 6= 0)isa(n): circle hyperbola point ellipse parabola pair of lines v. The trace in the plane z =0isa(n): circle hyperbola point ellipse parabola pair of lines vi. The trace in the plane z = k (k 6= 0)isa(n): circle hyperbola point ellipse parabola pair of lines (b) Identify the surface X. cone elliptic paraboloid hyperboloid of one sheet ellipsoid hyperbolic paraboloid hyperboloid of two sheets (c) True or False? The path described by the vector function r(t) = t, 1 2 t, 1 p2 t lies on X.

10 3. (13 points) Consider the two curves given by the position vector functions r 1 (t) =ht 2 +6t, 12 t 3 i and r 2 (u) =h2u 6, 4i (a) Find the equation of the tangent line to the curve given by r 1 (t) att =1. (Give your final answer into the form y = mx + b) (b) Find a vector v = hv 1,v 2 i that has length 7 and is orthogonal to the tangent vector to r 2 (u) at u =4. (c) The two curves have one point of intersection. Find the (acute) angle of intersection between the curves at this point. (Round your final answer to the nearest degree).

11 4. (13 points) You are sitting at the origin on the surface 4z x 2 y 2 = 0. You launch a water balloon into the air and its position at time t seconds is given roughly by the vector function r(t) =ht, 2t, 20t 5t 2 i. (a) Give the two word name of this surface. (b) Your math instructor just happens to be sitting at the location where the water balloon lands on the surface. Find the (x, y, z) locationwhereyourmathinstructorsissitting. (c) Find parametric equations for the tangent line to the path at t =2. (d) Find the curvature at time t =2.

12 Math 126C First Midterm Spring (9 points) At what point do the curves in R 3 intersect? r 1 (t) = t 1, 3t, t 2 and r 2 (t) = t +2, 1 t, t 3 +9 Find their angle of intersection, correct to the nearest degree.

13 Exercise 3 (10 points). For the curve r(t)=(3sin(2t),cos(4t)), find the tangent line at t = 8 π its parametric equations. What is the slope of this tangent line? and give 4

14 Math 126 Spring (9 points) (a) Use traces or reduce the equation to one of the standard forms to identify the surface given by the equation x 2 4x +2y 2 4y 4z =2. (b) Find a vector v of length 2 that is parallel to the tangent vector to at the point (10, 0, 1). r(t) =h4t 2 3t, 7sin t, e t2 4 i

15 3. (a) (4 points) The curve traced by the vector function r(t) =h2 sin(t), t, 3 cos(t)i is contained in three of the four surfaces sketched below. Write down the equations of the three surfaces which DO contain the curve and put an X under the surface which does NOT contain the curve. 5

16 (b) (4 points) Identify the surface given by the equation 4x 2 24x + 100y 2 25z 2 50z = 89. Use the terminology of surfaces from Section Either sketch the surface or descrobe its orientation. For example, write x 2 + y 2 =5z 5 is an elliptic paraboloid which opens up in the positive z direction with its lowest point at (0,0,1) or sketch. 6

17 4. Given the curve x = cos(2 t) y =sin(2 t) z =3 t 3/2, (a) (5 points) Find the parametric equations of the tangent line to the curve at the point when t = 4. (b) (5 points) Compute the length of the curve from the point where t = 0 to the point where t = 8. 7