Practice problems 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b. 1, 1 = c 1 3, 2 + c 2 2, 1. Solve c 1, c 2. 2. Suppose a is a vector in the plane. If the component of the a in the direction of 1, 1 is 2 and the component of a in the direction of 2, 1 is 7 5/5. Find a vector with length 3 that points in the opposite direction as a does. Assume a = x, y. Then, using the two components, you can write out two equations for x and y. Find x and y. the vector desired is 3a/ a. 3. Given a = 1, 1 and b = 2, 3, decompose b into two parts, such that one is parallel with a and one is perpendicular with a. b = b a a a a. b = b b = b (b a)a. a 2 4. Suppose we know a b = 2, a = 2, b = 3, (a b) c = 3. Find the following quantities: (a). The angle between a + b and a b. (b). [(a 2b) (b 2c)] c 5. Let A(5, 2, 3), B(6, 4, 0), C(7, 5, 1) and D(14, 14, 18). (1). Compute the area of the triangle ABC. (2). Compute the volume of the parallelepiped determined by AB, AC and AD. Is the frame given by { AB, AC, AD} left-handed or right handed? (3). Find the distance from D to plane ABC (You can just use the results from (1) and (2)) (1). 1 2 AB AC (2). V = ( AB AC) AD = determinant (3). d = V/ AB AC 6. Given two planes x + y + z = 5 and 3x y = 4, write a parametric equation and a symmetric equation for the line of intersection of the two planes. 7. Find the line that passes through (2, 6, 3) and parallel with both 3x 9y + 4z = 10 and 2x y + 4z = 12. 1
The direction of the line is parallel with the cross product of the normal vectors of the planes (why?). 8. Consider L 1 : x(t) = 3 2t, 4t 1, 3t+2 and L 2 : x(t) = 6t+3, 1 t, 2 2t. Are they skew, parallel or intersecting? If they are parallel or intersecting, find the plane that contains both lines. 9. (1). Consider the curve r(t) = 3t sin(t), 3t cos(t), 2t 2. Compute the arclength of this curve from t = 1 to t = 2. (2) Suppose that T (t) is the unit tangent vector of a curve. Explain why dt /ds is perpendicular with T where s is the arclength. (1). s = 2 1 r (t) dt (2). Use the fact that T is a constant. 10. Consider r(t) = t, t 2, t 3. Find the tangent line of this curve at (1, 1, 1) and a plane that is normal to this curve at (1, 1, 1). A vector tangent to the curve is the velocity v(t) = r (t). For the line, the line passes through (1, 1, 1) and is parallel with v(1); for the plane, it passes thorough (1, 1, 1) and is perpendicular with v(1). 11. (1). Describe the surface y 2 = x 2 + z 2 and get a rough sketch. (Hint: This is a surface of revolution.) (2). Find the trace of the ellipsoid x 2 /a 2 + y 2 /b 2 + z 2 /c 2 = 1, which is tangent to planes P 1 : x = 2, P 2 : y = 3 and P 3 : z = 1, in the plane that passes through (0, 0, 0) and is perpendicular with (1, 1, 0). (1). This is the cone revoluted from y = ±x, z = 0 about y-axis. (2). a = 2, b = 3, c = 1. The plane is simply x y = 0. hence, the trace is x2 4 + x2 9 + z2 1 = 1, y = x. This is an ellipse. *************************************************************** 1. Sketch the level curves of f(x, y) = x 2 y 2. The level curves are x 2 y 2 = k. Discuss k > 0, k = 0, k < 0 2. Can we define a suitable value for the function f(x, y) at (0, 0) to make it continuous? If yes, compute the value; if not, explain why. 2
(a). f(x, y) = xy x 2 +y 2 (b). f(x, y) = arctan( 1 x 2 +2y 2 ) (c). f(x, y) = x2 +y 6 y 2 +x 6 The key idea is to check if the limits exist as (x, y) (0, 0). If the limits exist, we can define the values to be the limits and then the functions become continuous. Part(a). The limit is zero as we showed in lecture. You can use polar coordinates. (b). The inside goes to positive infinity but the arctan then goes to π/2. The third one, by checking different directions y = kx. You see different limits. f(h,0) f(0,0) (1). Use the definition f x = lim h 0 h. (2). This is simply asking if f(x, y) f(0, 0) as you approach the origin. 3. Let P = ln x + y 2 and Q = 2xy + y 1 y 2. Can you find a function f(x, y) such that f x = P and f y = Q for x > 0, y < 1? If yes, find f; if not, explain why. This problem tests f xy = f yx. You should check that P y, Q x are continuous. They verify that if they are equal or not. 4. Suppose we want to compute (1.03) 1/3 ln(0.95) without calculator. Find a function f(x, y) and (x 0, y 0 ) such that f(x 0, y 0 ) = (1.03) 1/3 ln(0.95), and also (x 0, y 0 ) is close to (a, b) where f(a, b) is easily found. Use linear approximation of f to compute this value approximately. 5. Suppose f(u, v) is a differentiable function of u and v. Let w = f(u, v) ln(1 + u) xv. Suppose u = (x + y) 1/2 and v = xy. Com- at (x, y) = (2, 2) using chain rule. pute w x 6. Suppose f(x, y) = ln(x 2 + y 2 ). (1). Compute the tangent plane of the graph at (1, 1, ln 2). Can you find a normal vector of this plane? (2). Compute the tangent line of the level set f(x, y) = ln(2). Can you find a normal vector of this line? (3). On the level set f(x, y) = ln(2), near the point (1, 1), y can be regarded as a function of x. Compute y (x) x=1. (1). The tangent plane of the graph is given by z = f(1, 1) + f x (1, 1)(x 1) + f y (1, 1)(y 1). 3
Another way is to regard it as the level set of F = f(x, y) z = 0. Use the technique for computing tangent planes of level sets to compute the tangent plane. (2). The tangent line can be computed using two understandings. The first is the linear approximation: { } f(1, 1)+f x (1, 1)(x 1)+f y (1, 1)(y 1) ln(2) = 0 f x (1, 1)(x 1)+f y (1, 1)(y 1) = 0. The second understanding is that f(1, 1) is a normal vector of the line. Then, f(1, 1) x 1, y 1 = 0 (3). This tests implicit differentiation. y (1) = fx(1,1) f y(1,1) 7. Let f(x, y, z) = ln(1 + x 2 + y 2 z 2 ). (1). Find the direction in which f(x, y, z) decreases the fastest at (1, 1, 1). (2). Compute the rate of change of f with respect to distance in the direction indicated by 2, 1, 4 at (1, 1, 1). (3). Let r(t) = t 2, t 2, 2t 2 1. Compute r(1) and r (1). Then, compute d dt f(r(t)) t=1 using what you got just now. Explain how this is related to the directional derivative in (2). (1). As we explained, f points the fastest increasing direction while f points the fastest decreasing direction. In our case, we need a unit normal vector. Hence, the unit vector has something to do with f. (2). This is just directional derivative: D u f, but you should find u first. (3). Chain rule: d dt f = f(r(1)) r (1). You only need to compute f,r(1) and r (1). This rate of change w.r.t. time is equal to the directional derivative times speed. The speed is v = r (1). If you multiply this with the directional derivative, you get the same thing. 8. (1). Suppose P V nrt = 0 where n and R are two constants. Show that P V T V T P = 1 (2). Let F (x, y, z) = ln(x 2 + y 2 ) e x z. Consider the level set F = ln 2 1. This determines an implicit function. Near point (1, 1, 1), compute x y. 4
This problem tests implicit differentiation. Let F = P V nrt. Then, we have a level set F (P, V, T ) = 0. The product is 1. P V = F V F P V T = F T F V T P = F P. F T You can verify it directly: P = nrt/v. Hence, P/ V = nrt/v 2. Similarly you can compute others. The product is 1. (2). Use the formula for implicit differentiation. 9. The cone z 2 = x 2 + y 2 and the plane 2x + 3y + 4z + 2 = 0 intersect in an ellipse. Find the tangent line of the ellipse at P (3, 4, 5). Find the plane that is normal to the ellipse at P (3, 4, 5). (This is essentially your homework problem) Think about the situation. Will the tangent line be tangent to the cone? Will the tangent lint be in the old plane? If your answers are both yes, then you know that the tangent line is perpendicular with the normal of the cone and the normal of the old plane. How do you construct the vector parallel with the line? Cross product! How do you compute the normal of the cone? Gradient! The direction of the tangent line is a normal of the new plane, the second part is easy. 10. Find a plane that is tangent to the paraboloid z = 2x 2 + 3y 2 and is parallel with 4x 3y z = 10. What is the distance between the plane you find to the plane 4x 3y z = 10? (This is also essentially your homework problem.) How do you compute the normal of the tangent plane? Gradient! Wait, gradient of what? is that the gradient of f(x, y) = 2x 2 + 3y 2? No! Gradient of F that has the surface as a level set... Hence, F = 4x, 6y, 1. What is the relation between this normal with the plane given? It s parallel with the normal of the plane: F 4, 3, 1. Figure out x, y, z. The distance of the planes? They are parallel. Find a point on one plane, compute the distance from this point to the other plane. 5
11. F (x, y, z) = xyz + x 2 2y 2 + z 3. Find the tangent plane of the level set F = 14 at P (5, 2, 3). Find u such that D u F is the largest at P (5, 2, 3). F = yz + 4x, xz 4y, xy + 3z 2 is the normal of the level set. x 0 = 5, y 0 = 2, z 0 = 3, compute F. The plane is easy. Just directional derivative. It is the largest if u is in the same direction as the gradient. 12. z = 2x 4 8xy + 2y 4. Is there a highest point on the graph? if yes, find it. Is there a lowest point on the graph? If yes, find it. There is no global max since the function value can be very large. There must be a min in the middle somewhere. The min must be a critical point. hence, find all critical points check which one is the one you want. Some people may use 2nd derivative test to check the critical points. The 2nd derivative test is only for local min and local max. It s true that the global min must a local min. Then, if there is only one local min, then it must be the one you want. However, if you have several, you must compute the function values. Hence, to make your life easy, you probably compute the function values directly. 13. Find all critical points of f(x, y) = x 3 + y 3 + 3xy and classify them. This is straightforward. f x = 3x 2 + 3y = 0 and f y = 3y 2 + 3x = 0. Hence, y = x 2 and x 4 + x = 0. x = 0 or x = 1. Use y = x 2 instead of y 2 = x to determine y. The latter will give you more points which are not critical points. 14. Find all critical points of f(x, y) = 6xy 2 2x 3 3y 4 and classify them. For (0, 0), the test fails. Check the behavior near it and convince yourself that it is a saddle point. This is an example in the book. Read the book. 6