Module 1

1. Which of the following is a necessary and sufficient condition for a function, f, to be continuous at the point c = 4? O lim f (I) does not exist. O lim f (z) = f(4). O f is defined on an open interval that contains a = 4. O lim f (z) exists. 2. Consider the function: f(z) 12 - 1 - 1 Which of the conditions of continuity are not met by f (x) at x = 1? O f (1) must be defined. O lim f(z) must exist. O lim f(x) = f(1). 3. Let f (z) = 12 - 9 I - 3 Is f (I) continuous at I = 3? O No, f (I) is not continuous at I = 3. Yes, f (z) is continuous at I = 3. 4. Suppose f(I) = 12 - 4x+3 12 - 51 + 6 Which statement describes the continuity of f (I) at a = 3? O f is continuous at I = 3. O f has a removable discontinuity at I = 3. Of has a jump (nonremovable) discontinuity at a = 3. f has an infinite discontinuity at i = 3. 5. Suppose f (I)= I+ 7, 1>0 Which statement describes the continuity of f at a = 0? O f is continuous at I = 0. O f has a removable discontinuity at I = 0. O f has a jump (nonremovable) discontinuity at I = 0. O f has an infinite discontinuity at a = 0. 6. Let f (i) = = + 2 for # -2 and f (I) = 1 for I = -2. Explain why the function is discontinuous at -2. Of(-2) is not defined O f(-2) is defined, but lim, f(I) does not exist. O f(-2) and lim, f (i) are both defined, but lim, f(z) # f(-2). O lim f (z) and lim, f (z) are both defined, but lim f(z) + lim f(I) 7. Let f(x) = sin(z) if z > T/4. Explain why the function is continuous on (-0o, co). O f is continuous on (-00, oo) because both sin(I) and cos(I) are continuous on (-00, oo), and the sum of continuous functions is continuous. f is continuous on (-oo, oo) because both sin(z) and cos(z) are continuous on (-00, oo), and the composition of continuous functions is continuous. f is continuous on (-00, oo) because both sin(I) and cos(I) are continuous on (-0o, oo), and sin(I) = cos(z) when I = TT /4. O f is continuous on (-00, oo) because both sin(z) and cos(z) are continuous on (-00, oo), and sin(z) 4 Find the values of m and b that make f (I) differentiable. O m = 8,b= -28 O m =8,b= -16 O m = 4,b= -12 O m = 4,b=0 9. Find the derivative of the function using the definition of derivative. g(t) = Preview will appear here... Enter math expression here 10. Find the derivative of the function using the definition of the derivative. k(I) =1-12 Enter the expression for the derivative only, not k'(x) =. Preview will appear here... Enter math expression here1. Find the derivative of f if f (I) = 2-50. Preview will appear here.. Enter math expression here 2. Differentiate the function F(y) = cy-6. Preview will appear here.. Enter math expression here 3. Ifh = 3k2 and k = (3t)2, then find af - Preview will appear here... Enter math expression here 4. Differentiate the function. D(t) = 1 + 16t2 (4t)3 Preview will appear here... Enter math expression here 5. Find the equation of the tangent line to the curve y? = 13 at the point (1, 1). Oy= 2+3 Oy=27-2 Oy= - 2+3 The tangent line does not exist at the point (1, 1). 6. Find the equation of the tangent line to the curvey = 3e* + I at the point where a = 0. Oy= 4x +3 Oy = 4x - 12 Oy=3x +3 Oy=1+3 7. Differentiate the function g(I) = ex+2 + 2. O g'(z) = 6742 O g'(z) = ez+2 + 2 O g'(z) = 2ez+2 8. Find the derivative of J(z) = 5ez+ O J'(z) = 5e# + 723/2 O J'(z) = 5ex - 22-7/2 O J'(z) = 5e# + V524 O J'(z) = 5ex - 2-7/5 9. Differentiate the function q(I) = 2log3(I) + In(I) C -2+1 O q'(z) = 213 + 1 I O q(1) = In 3 + 1 O 10. Find the derivative of f (I) = log10 (100z). O f'(z) = 100 I In 10 O f'(x) = 100x In 10 O f' (I) = In 100 O I In 101. Find the derivative of f (x) = 3 sin . Preview will appear here... Enter math expression here 2. What is the derivative of g (x) = = sin(I)? Preview will appear here... Enter math expression here 3. Find the derivative of f(t) = tcostcsct Of'(t) = -tescot Of'(t) = - cott Of'(t) = cott -tcscot O f'(t) = t - cot2t 4. Find the 65th derivative of f (0) = sin 0. O - cos 0 cos 0 O sin 0 O - sin e 5. Differentiate g(0) = e (tan e - 0). O g'(0) = e(sec2 0 - 1) O g'(0) = e (tan 0 - 0 - csc2 0 - 1) O g'(8) = -e (csc2 0 + 1) O g'(0) = e (tan 0 - 0 + sec2 0 -1) a y ify = 6. Find COS I 1 - sin I O dy sinx - sin x + costa (1 - sin x)2 ay _ tan I O dy dx (1 - sin x)2 O dy 1 - sin x 7. Find an equation of the tangent line to the curve = e cost at the point (0, 1). Oy= -1+1 Oy=z+1 Oy= 2x +1 Oy= 1 8. For which values of T does the curve y = e sin c have a horizontal tangent line? ONone of these O z = -" + nx, n = 0,41, 42, ... O z = = + nx, n = 0, 41, 12, ... O z = " + nn, n = 0, 41, 12, ... O z= -7+ 2nm, n = 0, 41, 42, ... 9. A mass is attached to the end of an elastic band, and the elastic band is then hung over a hook. When the mass is pulled downward and released, it vibrates vertically. Its distance in centimeters below the hook after t seconds is given by p(t) = 4 cost + 2 sint. (Take the positive direction to be downward). Find the velocity v(t) and acceleration a(t) at time t. Select one answer for each. O v(t) = -4cost - 2 sint O v(t) = -4 sint + 2 cost O v(t) = 4sint - 2 cost O v(t) = 4cost + 2sint O a(t) = -4cost - 2 sint O a(t) = -4sint + 2 cost O a(t) = 4sint - 2cost O a(t) = 4cost + 2 sint 10. Consider the same situation from the previous question: A mass is attached to the end of an elastic band, and the elastic band is then hung over a hook. When the mass is pulled downward and released, it vibrates vertically. Its distance in centimeters below the hook after t seconds is given by p(t) = 4 cost + 2 sin t. (Take the positive direction to be downward). In what direction is the mass moving at time t = " ? The mass is moving upward. The mass is moving downward. Not enough information.1. Let f(x, y, z) = yet sin z. Find the gradient vector for f. O Vf(z, y, z) = (yet sin z, et sin z, yet cos z) O Vf(z, y, z) = (y sin z, et cos z, yet) O Vf(z, y, z) = (ye cos z, e sina, ye' sin z) O Vf(z, y, z) = (yet cos z, e cos z, yet sin z) None of these 2. Letg(z, y) = . Compute Vg(3, 1). Enter your answer as a linear combination of the standard basis vectors i and j . For example, the vector (2, 3) = 2 i + 3 j should be entered as 2*i+3*j Preview will appear here.. Enter math expression here 3. True or false? if f and g are differentiable real-valued functions, then V(fg) = fVg + gVf. True False Compute the directional derivative of g(I, y) = et sin y at the point (0, 0) in the direction u = Enter the exact value of the directional derivative - do not round to a decimal. Preview will appear here.. Enter math expression here 5. Find the directional derivative of f (x, y) = xy + x y" at the point (-1, 2) in the direction of the unit vector given by the angle 0 = Enter the exact value of the directional derivative - do not round to a decimal. Preview will appear here... Enter math expression here 6. What is the maximum rate of change of h(I, y, z) = zeTy at the point (-1, 0, 2)? Enter the exact value - do not round to a decimal. Preview will appear here. Enter math expression here 7. If f is a differentiable multivariable function, in which direction does f decrease most rapidly? Cannot be determined OVf(I) (I) JA - O OVf(-I) A hiker is climbing a hill whose shape is modeled by the surface z = 400 - yz 100 1000, where c and y are measured in feet. The hiker stops to rest at the point (100, 300, 210). Find the direction that creates the steepest ascent from this point. None of these O (-200, -90) 0 (-2,-3) (200, 90) 9. As before, a hiker is at the point (100, 300, 210) on a hill whose shape is modeled by the surface z = 400 - - 100 1000 Where I and y are measured in feet. The positive I-axis points east and the positive y-axis points north. If the hiker moves northwest, will they start to ascend or descend? The hiker will start to ascend. The hiker will start to descend. The hiker will stay at the same altitude. 10. As before, a hiker is at the point (100, 300, 210) on a hill whose shape is modeled by the surface z = 400 - 100 1000 where I and y are measured in feet. The positive T-axis points east and the positive y-axis points north. Which direction(s) results in the hiker staying at the same height? Select all that apply. 3 10 V109' V109/ 0 /10 3 V109' 109 O 10 3 V109' V109 O 3 10 V109' V109/1. Find the extreme values of f (x, y) = x2 + y' subject to the constraint ry = 1. Absolute maximum value: 0 Absolute maximum value: 1 Absolute maximum value: 2 No maximum value Absolute minimum value: 0 Absolute minimum value: 1 Absolute minimum value: 2 No minimum value 2. Find the absolute maximum and minimum values of f (x, y, z) = x - y + z subject to the constraint x2 + y2 + 22 = 3. Absolute maximum value: 0 Absolute maximum value: 2 Absolute maximum value: 3 No maximum value Absolute minimum value: -3 Absolute minimum value: -2 Absolute minimum value: 0 No minimum value 3. Find the absolute maximum and minimum values of the function f(x, y) = x2 + xy + y on the unit disk D - {(x, y/) | 22 + y's 1} Absolute maximum value: 0 Absolute maximum value: Absolute maximum value: 1 Absolute maximum value: 2 Absolute minimum value: -1 Absolute minimum value: 0 Absolute minimum value: 2 Absolute minimum value: 1 What is the absolute maximum value of the function f (x, y) = 2x + 3y on the region x2 + 4y? 0 O a, b> o O a > 0 and b 0, then the graph of the function is concave up, and if fiz