Math Exam PDF

# Math Exam ## Page 1 **Name:** Gabby Nowak **Person Number:** 50509063 This exam is worth a total of 100 points. You must show all work/steps for all questions. Correct answers without supported work will receive 0 credits. No cell phones are permitted in this exam. Any academic integrity violat...

# Math Exam ## Page 1 **Name:** Gabby Nowak **Person Number:** 50509063 This exam is worth a total of 100 points. You must show all work/steps for all questions. Correct answers without supported work will receive 0 credits. No cell phones are permitted in this exam. Any academic integrity violation will automatically result in a final letter grade of “F” in the course. 1. Let C(x) represent the total cost (in dollars) of manufacturing x iPhones. For each of the following parts, you must clearly state the specific units and write your answer in a complete sentence. a. Interpret the statement C(200) = 100,000. (4 points) It will cost $100,000 to make 200 iPhones. b. Interpret the statement C' (200) = 500. (6 points) It costs $500 to make each iPhone. c. Approximate the cost of making 203 iPhones. Interpret your answer in a sentence. You must show all work/steps. (6 points) $100,000 + $500 + $500 = $101,500 It will cost $101,500 to make 203 iPhones. ## Page 2 **Name:** Gabby Nowak **Person Number:** 50509063 2. Using techniques from Chapter 3, evaluate each of the following limits. Other methods of evaluation, including L'Hopital's Rule, even if correct, will receive NO CREDIT. You must show all work/steps and clearly justify your solution method. Failure to use proper limit notation will result in a deduction points. a. Let f(x) = * 2 if x ≥ 7 * √x + 9 if x < 7 * x² - 1 1. lim f(x) (3 points) x→7- (7)² - 1 = 48 2. lim f(x) (3 points) x→7+ 2 2 3. lim f(x) Be sure to justify your answer. (3 points) x→7 2 2 b. lim (2x + 3) / (4x⁵ - 1) (4 points) x→∞ lim (2/∞ + 3 / ∞) x→∞ 0 c. lim (4x⁵ - 1) / (7x⁵ + 1) (6 points) x→∞ lim (4/∞ - 1/∞) x→∞ (7/∞ + 1/∞) 4/7 ## Page 3 **Name:** Gabby Nowak **Person Number:** 50509063 3. Thirty thousand dollars is invested at 7.27% interest compounded continuously. When will the investment be worth $90,000? Leave your final answer as an exact value and label your units. You do NOT need to put your final answer into a calculator and provide a decimal value. (13 points) 90000 = 30000 * e^(0.0727)(t) 60000 = e^(0.0727)(t) ln 60000 = (0.0727)(t) ln 60000 / 0.0727 = t ~ 171.3 years ## Page 4 **Name:** Gabby Nowak **Person Number:** 50509063 4. Using techniques from Chapter 3, evaluate each of the following limits. Other methods of evaluation, including L'Hopital's Rule, even if correct, will receive NO CREDIT. You must show all work/steps and clearly justify your solution method. Failure to use proper limit notation will result in a deduction points. a. lim (x² + 1) / (x + 3) (5 points) x→-3 lim (x² + 1) x→-3 (x + 3) ((-3)² + 1) / (-3 + 3) 10 / 0 Undefined b. lim (x - 3) / (3x² + x - 12) (6 points) x→3 lim (x - 3) x→3 (3x² + x - 12) lim (x - 3) x→3 (x + 4)(3x - 3) lim 1 x→3 (x + 4) 1 / (3 + 4) = 1/7 c. lim (√x - 8) / (x - 64) (8 points) x→64 lim (√x - 8) x→64 (x - 64) lim (√x - 8) * (√x + 8) x→64 (x - 64) * (√x + 8) lim (x - 64) / (x - 64) * (√x + 8) x→64 lim 1 / (√x + 8) x→64 1 / (√64 + 8) = 1/16 ## Page 5 **Name:** Gabby Nowak **Person Number:** 50509063 5. For this question, you will find the equation of the tangent line to the graph of f at the given point. f(x) = x² - 3x + 2, (-1, 6) a. Using f'(x) = lim (f(x + h) - f(x)) / h, find the slope of the tangent line at the given point. You must use f'(x) = lim (f(x + h) - f(x)) / h; other methods, even if correct, will receive NO CREDIT. (14 points) lim ((x + h)² - 3(x + h) + 2 - (x² - 3x + 2)) / h h→0 lim (x² + 2xh + h² - 3x - 3h + 2 -x² + 3x - 2) / h h→0 lim (h² + 2xh - 3h) / h h→0 lim h + 2x - 3 h→0 (6) + 2x - 3 = 2x - 3 b. Write the equation of the tangent line in slope-intercept form. (4 points) y - y1 = m(x - x1) -12 (-1, 6) y - 6 = -1(x - (-1)) y - 6 = -1x - 1 y = -1x + 5 ## Page 6 **Name:** Gabby Nowak **Person Number:** 50509063 6. Use the graph f(x) to find the indicated limits and function values, or state that the limit or the function does not exist. Simply write the answer in the box; no work is required for this question. (1 point each, 15 points) | | Answer | |---|---| | a. lim f(x) as x→ -4+ | 1 | | b. lim f(x) as x→ -4- | 2 | | c. lim f(x) as x→ -4 | DNE | | d. f(-4) | DNE | | e. Is f continuous at -4? | No | | f. lim f(x) as x→ 1- | -2 | | g. lim f(x) as x→ 1+ | 4 | | h. lim f(x) as x→ 1 | DNE | | i. f(1) | DNE | | j. Is f continuous at 1? | No | | k. lim f(x) as x→ 6 | 2 | | l. Is f continuous at 6? | Yes | | m. lim f(x) as x→ 6+ | 5 | | n. lim f(x) as x→ 6- | 5 | | o. f(6) | 2 |