Math 105 Final Exam

University:
University of Michigan
Course:
MATH 105 | Data, Functions, and Graphs
Academic year:

2024
Views:

142

Pages:

2
Author:

Angeline Kim

Math 105 Final Exam 1. Evaluate each of the following limits. Justify your answers completely and be clear whether the limit equals a value, +1 or 1, or does not exist. x2 + 2x 3 x!2 x2 + x 2 2 x + 3x 4 (b) lim x!1 |1 x| (a) lim [5] [5] x2 x!2+ x2 7 x (d) lim 3x + 2 4x + 4 1 x+6 x! 7 x+7 x2 8x + 15 (e) lim , where g(x) = x2 + 7. 8x + g(x + 1) x!3 1 (c) lim [5] [5] [5] x2 4x 21 p x! 3 1 x 2 p x2 + 1 (g) lim x!1 2x 3 2 x 7x + 4 (h) lim 4 x! 1 x + x2 3 (f) lim [5] [5] [5] 2. Compute each of the following derivatives. x2 + 1 (a) f 0 (1), where f (x) = p . x x + 2x + 1 x (b) g 00 (x), where g(x) = 2 x dy (c) , where x2 y 3 + 3x3 2y 2 = 26. dx ✓ ◆4 ✓ 1 0 (d) g (x), where g(x) = + 7x x4 x3 p 3. Let f (x) = 3x 2. [5] [5] [5] 1 x7 ◆ 5 . [5] (a) Calculate f 0 (x), using the limit definition of the derivative. (b) Check your answer using the Chain Rule. 4. Consider the equation 2y + y 2 this curve at the point (1, 0). y 5 = x4 [5] 2x3 + 2x2 . Find the equation of the tangent line to 5. Find the absolute maximum and absolute minimum values of p f (x) = x2 5 6. Let f (x) = x on 2(x + 3) x4 and Page 1 of 2 [10] [15] [1, 5]. x+2 . Take my word for it that: x3 f 0 (x) = [10] [20] f 00 (x) = 6(x + 4) . x5 Math 105 Final Exam Analyze the domain of f , horizontal and vertical asymptotes, local extrema, and inflection points on the graph, as well as where the graph is increasing, decreasing, concave up and concave down. [Note: You do not need to compute y-values of extrema or inflection points.] Then use this information to present a detailed and labelled sketch of the curve y = f (x). x+2 6. (continued, if you need more space): Recall: f (x) = , x3 f 0 (x) = 2(x + 3) x4 f 00 (x) = and 6(x + 4) . x5 7. A water tank is in the shape of an inverted cone (i.e., circular opening at the top, vertex at the bottom). The diameter at the top is 12 feet, and the tank is 12 feet deep. Water is leaking out the bottom of the tank. When the water is 4 feet deep, the water level is dropping at a rate of 2 feet per minute. How fast is the water leaking out of the tank at that time? [20] 8. A large box with a square base and top is to be made from cardboard to hold a fixed volume of 54 cubic feet. Determine the dimensions that minimize the amount of cardboard needed. [20] 9. Consider the function 8 > <2 x f (x) = x + 4 > : (x 2)2 if x < 1 if 1 1 (a) Carefully sketch the graph of f (x). (b) State the domain of the function f (x). Recall 8 > <2 x f (x) = x + 4 > : (x 2)2 [10] [2] if x < 1 if 1 1 (c) Compute lim f (x) and lim f (x). x! 1 [10] x!1 (d) State the value(s) of x at which f is discontinuous. Justify your answer(s) using the definition of continuity. 10. Find and classify (as local maximum, local minimum, or neither) all critical numbers of g(x) = (x + 1)4 (x Page 2 of 2 2)3 . [3] [15]