Exam 2 Summary Disclaimer: The exam 2 covers lectures 9-15, inclusive. This is mostly about limits, continuity and differentiation of functions of 2 and 3 variables, and some applications. The complete ideas, definitions and theorems are included in the lecture notes and textbook. This is my attempt to help you study, and might not include everything that you need to know. Lecture 9: Functions of Several Variables 1. The domain of a function is the set of all possible inputes of the function and the range is the set of all outputs. 2. The domain of a function of 2 variables is a subset of R 2 and the domain of a function of 3 variables is a subset of R 3. In either case, the range is a subset of R. 3. The domain restrictions you learnt in Calculus I/ precalculus still apply. These include, but not limited to Domain restriction You must solve for 0 something 0 ln( ) > 0 (2k 1)π tan( ) 2 sin 1 ( ), cos 1 ( ) 1 1 Polynomials, exponential function and inverse tangent have no domain restrictions. 4. To find the range of a given function, freeze all but one variable (at meaningful values) to make your function a function of a single variable and investigate the range. Repeat this with different values/ other variables if necessary. Also keep in mind that the domain restrictions might affect the range. The following table of range of some standard functions from calculus I can be helpful. Function Range { x n (, ) if n is odd [0, ) if n is even e x (0, ) ln(x), tan(x) (, ) sin(x), cos(x) [ 1, 1] tan 1 (x) ( π/2, π/2) sin 1 (x) [ π/2, π/2] cos 1 (x) [0, π] 1
5. The level curves of a function f(x, y) of 2 variables are the curves you get by setting f(x, y) =Constant. The shape of the curve depends on the constant you choose. The question Characterize all level curves asks you to find all possible different shaped level curves obtained by using different constants. 6. The level surfaces of a function f(x, y, z) of 3 variables are the surfaces you get by setting f(x, y, z) =Constant. The shape of the curve depends on the constant you choose. The question Characterize all level surfaces asks you to find all possible different shaped level surfaces obtained by using different constants. In lecture 5 you studied some commonly occurring surfaces. Some of these equations are given below. Equation Surface ( x ) 2 ( y ) 2 + = z 2 Elliptic cone (Cone when a = b = 1) a b ( x ) 2 ( y ) 2 ( z ) 2 + = + 1 Hyperboloid (1 sheet) ( a b c x ) 2 ( y ) 2 ( z ) 2 + = 1 Hyperboloid (2 sheets) a b c z = x 2 + y 2 paraboloid z = x 2 y 2 Paraboloid hyperbolic ( x ) 2 ( y ) 2 ( z ) 2 + + = 1 Ellipsoid a b c Lecture 10: Limits of Multivariable functions 1. We say the limit of the function f(x, y) as (x, y) goes to (a, b) is L if the distance between f(x, y) and L can be made arbitrarily small by making the distance between (x, y) and (a, b) sufficiently close. 2. The above definition implies that the limit of the function at (a, b) does not depend on the path you take to reach the point (a, b). Which also implies that if the calculation of limit along two different paths gives two different values, then the limit at that point does not exist. 3. A function f(x, y) is said to be continuous at a point (a, b) if lim f(x, y) = f(a, b). (x,y) (a,b) This implies (a) f is defined at the point (a, b) (b) the limit lim f(x, y) exists. (x,y) (a,b) (c) They have the same numerical value. If any one of the above conditions fail, then the function is not continuous at the point (a, b). 4. If the function is continuous, you can plug in the value and find the limit. 2
5. If you are given a limit question (a) First plug in and see whether you get a finite value. If it does, then this is the limit value. If you get an infinite form (like 1/0) then the limit is infinite, and may not exist. (b) If you get an indeterminate when you plug in, see whether you can employ a calc I trick like Factor and cancel Multiply by the conjugate Clear the denominator using LCD to find the limit. (c) If none of the above works, see whether you can get 2 paths that goes to the given point, and gives you 2 different values when you calculate the limit. In this case, the limit does not exist. The ideas here can be extended naturally to a function of 3 variables by making the obvious but significant changes. Partial Derivatives of Multivariable func- Lecture 11: tions 1. The partial derivative of a function of two variables f(x, y) with respect to x at the point (a, b) is defined by the limit f(x, b) f(a, b) lim x a x a := f x (a, b) and the partial derivative of f(x, y) with respect to y at (a, b) is the limit f(a, y) f(a, b) lim y b y b := f y (a, b) Partial derivatives can be denoted in several types of notations. eg: f x (a, b) = f x = D x f(a, b) (a,b) 2. The partial derivative can be computed by thinking of all but the variable in question to be constants. 3. The partial derivative f x (a, b) gives the rate of change of the function in the direction of x, and f y (a, b) gives the rate of change of f in the direction of y. 4. Higher order partial derivatives can be obtained by successively differentiating the function partially with respect to the given order. eg: f xyx 2(a, b) means you need to partially differentiate f with respect to x, then with respect to y, and then twice with respect to x. 3
5. Clairaut s Theorem for mixed partial derivatives: Assume f is defined on an open set D in R 2 and both f xy and f yx are continuous through D. Then f xy = f yx for all (x, y) in D. Lecture 12: Tangent Planes, Linear Approximations and Differentials. 1. The Linearization of f(a, y) at the point (a, b) is the linear function L(x, y) = f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) This function can be used to approximate the values of f for points near (a, b). 2. Let f(x, y) 2 variable function f defined in a disk containing (a, b) is and let f x (a, b) and f y (a, b) exist. We say f is differentiable at (a, b) if it can be expressed in the form f(x, y) = L(x, y) + e(x, y) Where e(x, y) satisfies lim (x,y) (a,b) e(x, y) (x a)2 + (y b) 2 = 0 Note that the existence of partial derivatives at the point (a, b) is not sufficient to determine whether f is differentiable at (a, b) 3. In this case, the tangent plane to f at the point (a, b) exists and its formula is given by z = f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) This is the formula for equation of a tangent line of an explicitly defined function. 4. Criterion for differentiability: Assume f has partial derivatives defined in an open set D containing (a, b) with both f x and f y continuous at (a, b). Then f is differentiable at the point (a, b). 5. If f is differentiable at the point (a, b), then f is continuous at the point (a, b). The converse is false. 6. Formulas for differentials for a differentiable function at a point (a, b). The difference in independent variables x and y are denoted by dx and dy respectively and dx = x a and dy = y b. z is the difference of the functions value between those two points, so z = f(x, y) f(a, b). The differential dz = L(x, y) f(a, b), where L(x, y) is the linearization function of f at (a, b). It also gives the equality dz = f x (a, b)(x a) + f y (a, b)(y b). At points (x, y) near (a, b), z dz so z f x (a, b)(x a) + f y (a, b)(y b). 4
Lecture 13: Vector Directional Derivative and the Gradient 1. The gradient vector of a function f(x, y) of 2 variables is given by f(x, y) = f x (x, y), f y (x, y) For a function g(x, y, z) of 3 variables, this becomes g(x, y, z) = g x (x, y, z), g y (x, y, z), g z (x, y, z) 2. Let f be a differentiable function at (a, b) and let û be a unit vector in the xy-plane. The directional derivative of f at the point (a, b) in the direction of û is given by the formula 3. Properties of f. Dûf(a, b) = f(a, b) û (a) f is a vector field. That is, every point on f(a, b), which is a surface gets its own vector f(a, b). (b) f points in the direction of the maximum increase of f (or the steepest ascent). f points to the direction of the maximum increase of f (or the steepest descent) (c) Any vector orthogonal to f points to a direction where f does not change. (d) f is orthogonal to the level curves f has the same properties in 3 variables, and the last property should be changed to: f is normal to the level surfaces of f. Lecture 14: The Chain Rule, Implicit Differentiation, the Gradient and the level curves 1. If z is a function of x and y, and x and y are in turn functions of an independent variable t, the chain rule states dz dt = z dx x dt + z dy y dt In this case, z is the dependent variable, x and y are intermediate variables and t is the independent variables. 2. If z is a function of x and y, and x and y are in turn functions of two independent variables s and t, then the chain rule states z t = z x x t + z y y t 5 z and s = z x x s + z y y s
3. In other cases, you can find the corresponding derivatives/partial derivatives by drawing a variable tree as described in class. 4. Computationally, it is easier to plug in and write the function in terms of the independent variables and compute the required (partial) derivatives. 5. If F (x, y) = 0 is an implicit function of x and y, then the derivative dy dx formula dy dx = F x F y is given by the 6. As noted in lecture 13, The gradient vector f(x, y) of a 2 variable function is orthogonal to the level curves, and the gradient vector g(x, y, z) of a 3 variable function is normal to the level surfaces. 7. Tangent planes for an implicit function F (x, y, z) = 0 at the point (a, b, c) is given by F (a, b, c) (x a), (y b), (z c) = 0 Lecture 15: Local Maximum and Minimum Values, Second Derivative Test, and the Extreme Value Theorem 1. A function f(x, y) has a local maximum value at (a, b) if f(a, b) f(x, y) for all (x, y) in the domain of f in some open disk containing (a, b). A function f(x, y) has a local minimum value at (a, b) if f(a, b) f(x, y) for all (x, y) in the domain of f in some open disk containing (a, b). 2. A point (a, b) in the domain of f is called a critical point if either (a) Both f x (a, b) = 0 and f y (a, b) = 0 or (b) One of f x (a, b) or f y (a, b) is not defined. 3. Fermat s Theorem If f has a local maximum or a minimum at the point (a, b), then (a, b) is a critical number of f. Remark: The converse of Fermat s theorem is not true. That is, there can be points where the partial derivatives are zero or undefined, yet fail to be a local max or min. eg: Saddle points. Remark: Fermat s Theorem also implies that if f is differentiable and f has a local maximum or a local minimum at a point (a, b), then both f x (a, b) = 0 and f y (a, b) = 0. 4. A function f has a saddle point at a critical point (a, b) if, in every open disk centered at (a, b) there are points (x 1, y 1 ) for which f(x 1, y 1 ) > f(a, b) and points (x 2, y 2 ) for which f(x 2, y 2 ) < f(a, b). 6
5. Second Derivative Test: Suppose that the second order partial derivatives are continuous throughout an open disk centered at the point (a, b) where f x (a, b) = f y (a, b) = 0. We can define a new function D(x, y) called the discriminant by the formula D(x, y) = f xx (x, y)f yy (x, y) (f xy (x, y)) 2 (a) If D(a, b) > 0 and f xx (a, b) < 0, then f has a local maximum at (a, b). (b) If D(a, b) > 0 and f xx (a, b) > 0, then f has a local minimum at (a, b). (c) If D(a, b) < 0, then f has a saddle point at (a, b). (d) If D(a, b) = 0, the second derivative test is inconclusive. 6. A point (a, b) is said to be an absolute maximum of a function f defined in a domain D if f(a, b) f(x, y) for all (x, y) D. A point (a, b) is said to be an absolute minimum if f(a, b) f(x, y) for all (x, y) D. 7. Extreme Value Theorem If a function f is continuous on a closed and bounded domain D, then f has both an absolute maximum and an absolute minimum on D. Remark: By Fermat s Theorem, the absolute maximum or the minimum occurs either at critical points or on the boundary. 8. To find absolute extrema in a closed and bounded region (a) First find partial derivatives to find critical numbers of f on the domain. Evaluate the function s value at the critical numbers. (b) Parametrize the boundary, and find the critical numbers by differentiating. Evaluate the function on the points you found. (c) Pick the absolute maximum and the minimum. Make sure they are in the domain. 7
L9-L15 Practice problems 1. Find the domain and the range of the following functions (a) f(x, y) = x 2 sin(y) (b) f(x, y) = ex y 2 (c) f(x, y, z) = (d) f(x, y) = sin(z) 1 + x2 + y 2 sin(y) 1 x 2 (e) f(x, y, z) = tan 1 (x) yz 2. Describe the level curves of the functions below. (a) f(x, y) = x 2 y 2 (b) f(x, y) = x 2 + y 2 (c) f(x, y) = x + y. 3. Describe the level surfaces of the following functions (a) f(x, y, z) = x 2 + y 2 + 4z 2 (b) f(x, y, x) = 16 x 2 y 2 z 2 (c) f(x, y, z) = x 2 + y 2 z 2 4. Show that the following limit of the function does not exist at the given point by evaluating them in two suitably chosen paths. x lim 3 y (x,y) (0,0) x 2 + y 2/3 5. Evaluate the limit using continuity (a) (b) e x2 e y2 lim (x,y) (1,1) x + y lim (x,y) (2,3) tan 1 (x 2 y) 6. Evaluate the following limit lim (x,y) (0,0) 6xy x2 + y 2 or show that it does not exist.
7. Evaluate the following limits or show that it does not exist. (a) (b) (c) (d) (e) (f) lim (x,y) (0,0) lim (x,y) (0,0) y 2 x2 4 xy x2 + y 2 (x + y + 2)e 1/(x2 +y 2 ) lim (x,y) (0,0) 2 ( π (x 2 lim tan + y 2 ) (x,y) (0,0) 4 lim (x,y) (0,0) x 2 + y 2 x2 + y 2 + 1 1 lim (1 + x)y/x (x,y) (0,0) ) tan 1 ( ) 1 x 2 + y 2 8. Compute all the first order partial derivatives of the following functions. (a) z = 4 x 2 y 2 (b) z = (sin(x))(sin(y)) (c) z = sin(u 2 v) (d) f(x, y, z) = x y + z (e) f(x, y) = x y 9. Find the equation of the tangent plane for the following functions at the given points. (a) f(x, y) = x 2 y + xy 3 at (2, 1) (b) f(x, y) = x y at (4, 4) (c) g(x, y) = e x/y at (2, 1) 10. Find points on z = xy 3 + 8y 1 where the tangent plane is parallel to 2x + 7y + 2z = 0. 11. Write the linear approximation to f(x, y) = x(1 + y) 1 at (a, b) = (8, 1). Then use it to estimate 7.9 2.1 12. Let I = W be the Body Mass Index (BMI). A boy has W = 65kg and height H = 1.5m. Measure the H2 change in I if (W, H) changes to (66, 1.55) 13. Calculate the directional derivative of the following functions in the direction of the vector u at the given point. (a) f(x, y) = x 2 y 3 in the direction of u = ı + 2j at the point ( 2, 1). (b) f(x, y) = sin(x y) in the direction of u = 1, 1 at the point (π/2, π/6). (c) f(x, y) = e xy y2 in the direction of u = 12, 5 at the point (2, 2). (d) g(x, y, z) = xe yz in the direction of u = 1, 1, 1 at the point (1, 2, 0). 14. Find the directional derivative of f(x, y) = x 2 + 4y 2 at the point P = (3, 2) in the direction pointing to the origin. 15. Suppose that for a point P, f p = 2, 4, 4. Is f increasing or decreasing in the direction of u = (2, 1, 3) 2
16. (a) Evaluate f if f(x, y) = xy 2. (b) What is f(2, 3)? (c) Find the unit vector that points in the direction of maximum change at (2, 3). ( ) x 1 17. Let f(x, y) = tan 1 1 and u = 2,. y 2 (a) Calculate the gradient of f (b) Calculate Dûf(1, 1) and Dûf( 3, 1). (c) Show that the lines y = mx for m 0 are level curves of f. 18. Use the chain rule to calculate the partial derivatives, express the answer in terms of the independent variable. (a) f(x, y, z) = xy + z 2. Where x = s 2, y = 2rs and z = r 2. Find f f and s r. (b) R(x, y) = (3x + 4y) 5. Where x = u 2 and y = uv. Find R u and R v. (c) f(x, y, z) = xy z 2. Where x = r cos(θ), y = cos 2 (θ) and z = r. Find f θ. (d) f(x, y, z) = x 3 + yz 3 where x = u 2 v, y = u + v 2 and z = uv. Find f u ( 1, 1) and f v ( 1, 1). 19. Let let z be given as an implicit function F (x, y, z) = x 2 z + y 2 z + xy 1 = 0. (a) Find F x, F y and F z. (b) Using (a), find z x and z y. (c) Find the equation of the tangent plane of the above given surface at the point (1, 0, 1). 20. Find the equation of the equation of the tangent line to the implicitly defined function x 2 + y 2 z 2 = 1 at the point (1, 1, 1). 21. Find all the critical points of f(x, y) = 8y 4 + x 2 + xy 3y 2 y 3 22. (a) Find the critical numbers of the function f(x, y) = y 2 x yx 2 + xy. (b) Use the second derivative test to determine the nature of the critical points. 23. For the following functions, first find all the critical numbers, and then determine the nature of the critical points using the second derivative test. (a) f(x, y) = x 2 + y 2 xy + x. (b) f(x, y) = 4x 3x 2 2xy 2 (c) f(x, y) = x 3 xy + y 3 (d) f(x, y) = e x xe y 24. Find the absolute extrema of the following functions on the given domain (a) f(x, y) = (x 2 + y 2 + 1) 1 on 0 x 3 and 0 y 5. (b) f(x, y) = e x2 y 2, x 2 + y 2 1. (c) f(x, y) = x + y x 2 y 2 xy on 0 x 2 and 0 y 2. 25. Find the point on the plane x x y = 1 closest to the point (1, 0, 0). 3
26. Find the maximum volume of the largest box of the type shown in the following figure, with one corner at the origin and the opposite cornere at a point P = (x, y, z) on the paraboloid with x, y, z 0. z = 1 x2 4 y2 9 4
MAC2313 Test 2A (5 pts) 1. The equation of the tangent plane to surface 25x 2 + 4y 2 + 4z 2 = 100 at the point (0, 4, 3) is: A. 32y + 24z = 200 B. 8x + 24y + 32z = 200 C. 8x + 32y + 24z = 200 D. 24y + 32z = 200 E. none of the above (5 pts) 2. If w is a function of the variables x, y, and z and each of the variables x, y, and z is a function of the variables s and t, then which of the following is true? A. w s = w x x s + w y y s + w z z s B. dw ds = w dx x ds + w dy y ds + w dz z ds C. dw ds = w x x s + w y y s + w z z s B. w s = w dx x ds + w dy y ds + w dz z ds E. none of the above (5 pts) 3. The derivative 2 x y (y + y2 e x ) is equal to: A. ye x B. 2ye x C. x + y + ye x D. 2x + 2y + 2ye x E. 1 + 2ye x (5 pts) 4. If u = 2/ 13, 3/ 13 and f(x, y) = x 2 + y, then D u f(1, 0) is equal to: A. 2/ 13 B. 3/ 13 C. 5/ 13 D. 6/ 13 E. 7/ 13
(5 pts) 5. Which of the following are critical points of the function f(x, y) = x 2 +2y 2 4xy 6x? I. (3, 3) II. ( 3, 3) III. (0, 0) IV. (1, 1) A. only I B. only II C. only III and IV D. only I and II E. only I, III, and IV (5 pts) 6. The equation of the tangent plane to the surface z = cos(xy) at the point (2, π, 1) is given by: A. L(x, y) = πx + 2y B. L(x, y) = πx + 2y 2π C. L(x, y) = πx + 2y + 2π D. L(x, y) = πx + 2y + 4π E. none of the above (5 pts) 7. Given the function g(x, y, z) = 16 x 2 y 2 z 2, which of the following are true? I. D = { (x, y, z) 16 x 2 + y 2 + z 2 } II. R = [0, 16] III. The level surfaces are spheres. IV. The graph of the function g(x, y, z) lies in R 4. A. only I B. only III C. only II and IV D. only I and II E. only I, III, and IV
(5 pts) 8. If f(x, y) = (2xy + 3y 2 ) 3, then f x ( 1, 1) is equal to: A. 4 B. 4 C. 6 D. 8 E. 12 (5 pts) 9. Let (a, b) be a point in the domain of the real-valued function f(x, y) and let L R; which of the following are true? I. If f(x, y) is continuous at (a, b), then lim (x,y) (a,b) f(x, y) exists. II. If lim (x,y) (a,b) f(x, y) exists, then f(x, y) is continuous at (a, b). III. If f(x, y) is continuous at (a, b), then lim (x,y) (a,b) f(x, y) = f(a, b). IV. If lim (x,y) (a,b) f(x, y) = L, then lim (x,y) (a,b) f(x, y) = L. A. only I B. only II C. only I and II D. only II and IV E. I and III (5 pts) 10. For f(x, y) = x 3 + y 3 12xy, the critical point (4, 4) is: A. a local max B. a local min C. a saddle point D. a boundary point E. the second derivative test fails at this point (5 pts) 11. To find the point on the surface z = x 2 y 2 closest to the point (1, 2, 3), which of the following functions should be optimized? A. x 2 + y 2 + z 2 B. (x + 1) 2 + (y + 2) 2 + (z + 3) 2 C. x 2 + y 2 + (x 2 y 2 ) 2 D. (x 1) 2 + (y 2) 2 + (x 2 y 2 3) 2 E. none of the above
(5 pts) 12. Which of the following is a vector orthogonal to the tangent line to the curve x 2 4xy y 2 = 4 at the point (5, 1)? A. 6, 2 B. 1, 3 C. 1, 3 D. 6, 22 E. 11, 3 (5 pts) 13. Let f(x, y, z) = x 2 + yz + cos(πz), x = t, y = cos t, and z = e t. Calculate the value of derivative of the f(x, y, z) with respect to t when t = 0. A. 0 B. 1 C. 1 D. 1 + π E. 1 π (5 pts) 14. Bonus. Let f(x, y) function which is differentiable at the point (a, b) and let u be a nonzero vector in R 2. Which of the following are true? I. The equation of the tangent plane to the surface at (a, b, f(a, b)) is given by L(x, y) = f y (a, b)(x a) + f x (a, b)(y b). II. D u f(a, b) = f x (a, b), f y (a, b) u III. The max rate of change of f(x, y) at (a, b) is in the direction of f(a, b). IV. If f(a, b) = 0, then (a, b) is a critical point of f(x, y). A. I and III B. II and III C. I and IV D. III and IV E. II and IV
MAC2313 Test 2A Name: UF-ID: Section: (6 pts) 1. Calculate the following limit: lim (x,y) (2,2) x 2 + y 2 x y. (6 pts) 2. Given f(x, y) = 2x 2 + y 2, use the value of the function at the point (2, 1) and differentials to approximate f(1.95, 1.1).
(7 pts) 3. Use the technique of implicit differentiation discussed in class to calculate dy dx given x2 y 2 + 3y 4 = 2e 3. (6 pts) 4. Given f(x, y) = x sin y + x 2, at the point (1, π) find a vector which points in the direction of no rate of change of the function.
(10 pts) 5. Given f(x, y) = (x 2 1)(y 2 4), f x (x, y) = 2x(y 2 4), and f y (x, y) = 2y(x 2 1), find the max and min values of f(x, y) on the rectangular region bounded by the lines x = 1, x = 1, y = 2, and y = 2.
MAC2313 Test 2 A (5 pts) 1. The equation of the tangent plane to the surface z = 2x 2 3xy + 4y 2 at (1, 2, 12) is given by: A. 2x 13y+z = 12 B. 2x 13y+z = 24 C. 2x+13y+z = 12 D. 2x+13y+z = 24 E. none of the above (5 pts) 2. Given a function z = f(x, y) with continuous first partial derivatives at (a, b), which of the following are true for (x, y) close to (a, b)? I. f(x, y) f x (a, b)(x a) + f y (a, b)(y b) II. z f x (a, b)(x a) + f y (a, b)(y b) III. f(x, y) dz + f(a, b) A. only I B. only II C. only III D. only I and III E. only II and III (5 pts) 3. If g(x, y, z) = xyz 2 + cos(zy) + e 2x 3y, what is the value of the derivative the point (1, 0, 2)? 3 g y 2 z at A. 4 B. 2 C. 0 D. 2 E. 4 (5 pts) 4. What is the domain of the function f(x, y) = x ln y 1 x 2 y 2? A. { (x, y) y 1, x 2 + y 2 < 1 } B. { (x, y) y 1, x 2 + y 2 > 1 } C. { (x, y) y 0, x 2 + y 2 1 } D. { (x, y) y > 0, x 2 + y 2 < 1 } E. { (x, y) y > 0, x 2 + y 2 1 }
(5 pts) 5. What is the range of the function f(x, y) = ln(x2 +y 2 ) 2+sin(x+y)? A. (, ) B. (0, ) C. [0, ) D. (, 0) E. (, 0] (5 pts) 6. The level curves of z 2 = 4x 2 + 4y 2, z > 0 are all circles. A. true B. false (5 pts) 7. A level curve is a trace generated by intersecting the graph of a function with a plane parallel to the x, y-coordinate plane. A. true B. false (5 pts) 8. Which of the following are true?: I. along the curve y = 0, lim (x,y) (0,0) (y 2 x)/(y + x) = 1 II. along the curve x = 0, lim (x,y) (0,0) (y 2 x)/(y + x) = 0 III. along the curve y = x, lim (x,y) (0,0) (y 2 x)/(y + x) = 1/2 IV. the lim (x,y) (0,0) (y 2 x)/(y + x) exists A. only I B. only II C. only III D. only I, II, and III E. I, II, III and IV
(5 pts) 9. Given a function f(x, y) such that f(a, b) 0 and f(a, b) = z 0, f(a, b) is parallel to the tangent line of the level curve f(x, y) = z 0 at (a, b). A. true B. false (5 pts) 10. Let f(x, y) be a function differentiable at the point (a, b) and let v = h, k be a unit vector; which of the following are true? I. D v f(a, b) > 0 II. D v f(a, b) = f(a, b) v III. the vector v points in the direction of the maximum rate of change of f at (a, b) IV. D v f(a, b) = f(a, b) sin θ where θ is the angle between v and f(a, b) A. only I B. only II C. only II and III D. only II and IV E. I, II, III, and IV (5 pts) 11. Given g(x, y, z) = 2x 2 3yz 3, g(1, 0, 1) is orthogonal to which of the following vectors: A. 3, 1, 2 B. 0, 2, 1 C 0, 2, 1 D. 0, 2, 0 E. 3, 4, 7 (5 pts) 12. If the Method of Lagrange Multipliers is used to find the maximum value of f(x, y, z) = 2x 2 + 3xy 2 + 4z 3 on the surface x 2 + 2y 2 + 5z 2 = 10, which of the following is not one of the equations which must be solved simultaneously: A. 12z 2 = 10λz B. x 2 + 2y 2 + 5z 2 = 10 C. 4x + 6xy = 2λx D. 6xy = 4λy
(5 pts) 13. Which of the following vectors point in the direction of the maximum rate of decrease of f(x, y) = e 2x+3y cos y at the point (0, 0): A. 3, 2 B. 2, 3 C 2, 3 D. 3, 2 E. none of the above (5 pts) 14. Bonus. Given f(x, y) = 2x 2 + 3y 4, if the linearization of this function at (0, 0) is used to approximate the function at (0.1, 0.2), what value is obtained? A. 4 B. 3.8 C. 3.6 D. 4.6 E. none of the above (5 pts) 15. Bonus. Let f(x, y) be a function with continuous second partial derivatives and let (a, b) be a point such that f x (a, b) = f y (a, b) = 0; using the second derivative test, which of the following are true? I If D(a, b) < 0 and f xx (a, b) > 0, then f has a local max value at (a, b). II. If D(a, b) = 0, then f has a saddle point at (a, b). III. If D(a, b) < 0 and f xx (a, b) < 0, then f has a local min value at (a, b). IV. D(a, b) = f xx (a, b)f yy (a, b) [f xy (a, b)] 2. A. only II B. only IV C. only I and III D. only I, III, and IV E. only I, II and III
MAC2313 Test 2A Name: UF-ID: Section: (7 pts) 1. If w = 2xz e y, x = s + t, y = s 3t, and z = st, write the chain rule for the derivative of w with respect to t; use the chain rule to calculate the derivative. (5 pts) 2. Find the critical point(s) of the function f(x, y) = x 2 (xy + 12) 8y.
(6 pts) 3. Find the vector equation of the tangent plane to the ellipsoid x 2 + 3y 2 + 4z 2 = 8 at the point (1, 1, 1). (7 pts) 4. Characterize the critical point (1, 1) of the function f(x, y) = x + x 2 y + y + xy 2.
(10 pts) 5. Use the Method of Lagrange Multipliers to find the maximum and minimum values of f(x, y) = 2x 6y 1 on the ellipse x 2 + 3y 2 = 1.
MAC2313 Test 2A (5 pts) 1. Let z = x 2 + ln y. How many of the following are true? i. The domain of the function is { (x, y) y 0 }. ii. The range of the function is [0, + ). iii. The given function is explicit. iv. The graph of the function is a surface in R 3. A. 0 B. 1 C. 2 D. 3 E. 4 (5 pts) 2. A level curve is a trace generated by intersecting a surface with a plane parallel to the y, z-coordinate plane. A. True B. False (5 pts) 3. Let f(x, y) = x2 y x 2. How many of the following are true? +y i. Along the curve y = x 2, lim (x,y) (0,0) f(x, y) = 0. ii. Along the line x = 0, lim (x,y) (0,0) f(x, y) = 1. iii. Along the line y = 0, lim (x,y) (0,0) f(x, y) = 1. iv. The limit, lim (x,y) (0,0) f(x, y), exists. A. 0 B. 1 C. 2 D. 3 E. 4
(5 pts) 4. If g(x, y, z) = z cos(x 2 z y 2 ) what is the value of 2 g x y when (x, y, z) = ( 2π, π, 1)? A. 0 B. 8π C. 4π D. 4 2π E. none of the above (5 pts) 5. The equation of the plane tangent to z = 3x 2 + 4y 2 when (x, y) = (1, 3) is given by: A. z = 8x + 18y 35 B. z = 4x + 24y 39 C. z = 6x + 24y + 39 D. z = 8x + 18y + 35 E. none of the above (5 pts) 6. Let z = 3x 3 y; what is the approximate value of the change of z when (x, y) changes from (2, 1) to (1.9, 1.05)? A. 0.64 B. 2.4 C. 0.46 D. 1.4 E. 2.8
(5 pts) 7. If f(x, y) = 5y sin x + y 5, what is D u f(0, 1) when u points in the direction of the vector 3, 4? A. 5 B. 0 C. 5 D. 5 E. none of the above (5 pts) 8. Consider the function f(x, y) = 2e x2 +3y at the point (x, y) = (0, 0); which of the following vectors gives the direction in which the function is not changing? A. 0, 6 B. 6, 0 C. 3, 4 D. 3, 1 E. 0, 6 (5 pts) 9. Given a function f(x, y) differentiable at a point (a, b), the line tangent to the level curve of f at (a, b) is orthogonal to the gradient f(a, b) provided f(a, b) 0. A. True B. False (5 pts) 10. The function f(x, y) = 2y + x 2 y 3xy has how many critical points? A. 0 B. 1 C. 2 D. 3 E. 4
(5 pts) 11. For f(x, y) = x 4 + y 4 4xy + 1, the critical point (0, 0) is: A. a local min B. a local max C. a saddle point D. a boundary point E. the second derivative test fails at this point (5 pts) 12. If we use the Method of Lagrange Multipliers to determine the max value of f(x, y, z) = 5x 3y + z on the ellipsoid x 2 + y 2 + 4z 2 = 10, how many of the following equations are included in the system which must be simultaneously solved? i. 1 = 8λz ii. x 2 + y 2 + 4z 2 = 0 iii. 3 = 2λy iv. 1 = 4λz A. 0 B. 1 C. 2 D. 3 E. 4
(5 pts) 13. Let f(x, y) be a twice differentiable function on a region D of the x, y-plane and let (a, b) be an interior point of D. How many of the following are true? i. The function is continuous at (a, b). ii. The discriminant of f(x, y) at (a, b) is equal to f xx (a, b)f yy (a, b) f xy (a, b). iii. It is possible to draw a sufficiently small circle surrounding (a, b) such that every point in the circle is in D. iv. The point (a, b) is a critical point of f(x, y). A. 0 B. 1 C. 2 D. 3 E. 4 (5 pts) 14. Bonus. The level surface 9x 2 + 7y 2 + 2z 2 = C, is defined for all real numbers C. A. True B. False
MAC2313 Test 2A Name: UF-ID: Section: (5 pts) 1. The function f is a function of the single variable t and in turn, t is a function of the variables u, v, and w; using the chain rule, give an expression for the derivative of the function f with respect to w. (8 pts) 2. The density of a thin circular plate of radius 3 is given by ρ(x, y) = 6 + xy and the edge of the plate is described by the parametric equations x = 3 cos t, y = 3 sin t, for 0 t 2π. Use the chain rule to find the rate of change of density with respect to t on the edge of the plate. Express your final answer in terms of t.
(10 pts) 3. Find and classify all of the critical points of the function f(x, y) = x 2 +6xy+3y 2 4y.
(12 pts) 4. Use the Method of Lagrange Multipliers to find the min value of f(x, y, z) = 2x 2 + y 2 + 2z 2 on the plane 2x + 3y + 4z = 19.