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Assignment 1 is due on Sunday, October 08, 2023 at 11:59pm. The number of attempts available for each question is noted beside the question. If you are having trouble figuring out your error, you should consult the textbook, or ask a fellow student, one of the TA’s or your professor for help. There are also other resources at your disposal, such as the Mathematics Continuous Tutorials. Don’t spend a lot of time guessing – it’s not very efficient or effective. Make sure to give lots of significant digits for (floating point) numerical answers. For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 2?3 instead of 8, sin(3 ? pi/2)instead of -1, e?(ln(2)) instead of 2, (2 + tan(3)) ? (4?sin(5))?6?7/8 instead of 27620.3413, etc. Problem 1. (1 point) Given that f(x) = x 6 h(x) h(?1) = 2 h 0 (?1) = 5 Calculate f 0 (?1) = Answer(s) submitted: • ?7 submitted: (correct) recorded: (correct) Problem 2. (1 point) Find g 0 (4) given that f(4) = ?3 and f 0 (4) = 2 (a) For g(x) = 3x 2 ?5 f(x) g 0 (4) = (b) For g(x) = 2x+1 f(x) g 0 (4) = Answer(s) submitted: • 14 • ? 8 3 submitted: (correct) recorded: (correct) Problem 3. (1 point) Find the following given that f(0) = 4, f 0 (0) = ?3, g(0) = 7,and g 0 (0) = ?7. (a) (f +g) 0 (0) = (b) (f ? g) 0 (0) = (b) (f /g) 0 (0) = Answer(s) submitted: • ?10 • ?49 • 1 7 submitted: (correct) recorded: (correct) Problem 4. (1 point) Find f 0 , given f(x) = x sin3 x 4 ? x 4 +1 f 0 (x) = Answer(s) submitted: • sin3 x 4 +12x 3 sin2 x 4 cos x 4 ? x 4 +1? x sin3 (x 4 )·2x 3 ? x 4+1 x 4 +1 submitted: (incorrect) recorded: (incorrect) Problem 5. (1 point) Find the limit lim h?0 f(?10+h)? f(?10) h , if f(x) = p3 943?6x 2 . lim h?0 f(?10+h)? f(?10) h = Answer(s) submitted: • 40 49 submitted: (correct) recorded: (correc Problem 6. (1 point) Find a and b so that the function f(x) = ( 6x 3 ?7x 2 +5, x < ?2, ax+b, x ? ?2 is both continuous and differentiable. a = b = Answer(s) submitted: • 100 • 129 submitted: (correct) recorded: (correct) Problem 7. (1 point) The tangent line to y = f(x) at (?10,?4) passes through the point (6,4). Compute the following. a.) f(?10) = b.) f 0 (?10) = Answer(s) submitted: • ?4 • 1 2 submitted: (correct) recorded: (correct) Problem 8. (1 point) Let f and g be functions such that f(6) = 30, f 0 (6) = 3, f(30) = 17, f 0 (30) = ?7, g(30) = ?29, g 0 (30) = ?10, g(17) = ?27,and g 0 (17) = 49. (a) (g ? f) 0 (30) = (b) (f ? f) 0 (6) = Answer(s) submitted: • ?343 • ?21 submitted: (correct) recorded: (correct) Problem 9. (1 point) Use linear approximation to approximate ? 81.4 as follows. Let f(x) = ? x. The equation of the tangent line to f(x) at x = 81 can be written in the form y = mx+b. Compute m and b. m = b = Using this find the approximation for ? 81.4. Answer: Answer(s) submitted: • 1 18 • 9 • 2 15 submitted: (score 0.333333333333333) recorded: (score 0.333333) Problem 10. (1 point) The acceleration due to gravity, g, is given by g = GM r 2 , where M is the mass of the Earth, r is the distance from the center of the Earth, and G is the uniform gravitational constant. (a) Suppose that we increase from our distance from the center of the Earth by a distance ?r = x. Use a linear approximation to find an approximation to the resulting change in g, as a fraction of the original acceleration: ?g ? g× (Your answer will be a function of x and r.) (b) Is this change positive or negative? ?g is [?/positive/negative] (Think about what this tells you about the acceleration due to gravity.) (c) What is the percentage change in g when moving from sea level to the top of Mount Elbert (a mountain over 14,000 feet tall in Colorado; in km, its height is 4.34 km; assume the radius of the Earth is 6400 km)? percent change = Answer(s) submitted: • no response • no response • no response submitted: (incorrect) recorded: (incorrect) 2 Problem 11. (1 point) Find the equations of the tangent and normal lines to the graph of the function at the given point. g(t

) = (4 sint ?3 cost) 2 at t = ? 2 . Tangent line: y = . Normal line: y = . Answer(s) submitted: • 2(4 sin(t)?3 cos(t))(4 cos(t) +3 sin(t)) • ? 1 24 t ? ? 2 +16 submitted: (score 0.5) recorded: (score 0.5) Problem 12. (1 point) Compute f 0 (x), f 00(x), f 000(x), and then state a formula for f (n) (x), when f(x) = ? 2 x f 0 (x) = f 00(x) = f 000(x) = f (n) (x) = [Hint: The expression (?1) n has value 1 if n is even and ?1 if n is odd. This expression can be used in your answer for the last part.] Answer(s) submitted: • 2x ?2 • ?4x ?3 • 12x ?4 • ?2n(?1) n x ?(n+1) submitted: (score 0.75) recorded: (score 0.75) Problem 13. (1 point) Let f(x) = 5x 2 cos(7x). f 0 (x) = f 0 (2) = f 00(x) = f 00(2) = Answer(s) submitted: • 10x cos(7x)?35x 2 sin(7x) • 20 cos(17)?140 sin(14) • 10 cos(7x)?245x 2 cos(7x)?140 sin(7x) • ?970 cos(14)?140 sin(14) submitted: (score 0.25) recorded: (score 0.25) Problem 14. (1 point) A particle moves along a straight line and its position at time t is given by s(t) = 2t 3 ?27t 2 +108t where s is measured in feet and t in seconds. Find the velocity (in ft/sec) of the particle at time t = 0: Answer: The particle stops moving (i.e. is in a rest) twice, once when t = A and again when t = B where A < B. Determine A and B. A = B = What is the position of the particle at time 18? Answer: Finally, what is the total distance the particle travels between time 0 and time 18? Answer: Answer(s) submitted: • 108 • 3 • 6 • 4860 • 4974 submitted: (score 0.8) recorded: (score 0.8) 3 Problem 16. (1 point) Use the differential to approximate the number sin(57? ). Answer: Answer(s) submitted: • no response submitted: (incorrect) recorded: (incorrect) Problem 17. (1 point) Find the local linear approximation of the function f(x) = ? 19+x at x0 = 6, and use it to approximate ? 24.9 and ? 25.1. (a) f(x) = ? 19+x ? (b) ? 24.9 ? (c) ? 25.1 ? For parts (b) and (c), you should enter your answer as a fraction. If you enter a decimal, make sure that it is correct to at least six decimal places. Answer(s) submitted: • no response • no response • no response submitted: (incorrect) recorded: (incorrect) Problem 18. (1 point) Use differentials to approximate the given values by hand. Provide 4 decimal digits of accuracy. (Note: Only use a calculator for routine arithmetic (addition, subtraction, multiplication, division) as part of the approximation by differentials. If you use your calculator to obtain a result directly (i.e., taking powers or roots), then your answer may be different than the requested approximation.) 8.072 ? 7.982 ? 2.043 ? 1.953 ? Answer(s) submitted: • no response • no response • no response • no response submitted: (incorrect) recorded: (incorrect) Problem 19. (1 point) The edge of a cube was found to be 50 cm with a possible error of 0.4 cm. Use differentials to estimate the maximum possible error in the calculated volume of the cube. Error ? cm3 Answer(s) submitted: • no response submitted: (incorrect) recorded: (incorrect) Problem 20. (1 point) Find the linearization L(x) of the function g(x) = x f(x 2 ) at x = 2 given the following information. f(2) = 1 f 0 (2) = 12 f(4) = 6 f 0 (4) = ?2 Answer: L(x) = Answer(s) submitted: • no response submitted: (incorrect) recorded: (incorrect) Don't use plagiarized sources. Get Your Custom Essay on Assignment 1 is due on Sunday, October 08, 2023 at 11:59pm.The number of attempts available for each question is noted beside the question. If you are having trouble figuring out your error, youshould consult the textbook, or ask a fellow student, one of the TA’s or your professor for help.There are also other resources at your disposal, such as the Mathematics Continuous Tutorials. Don’t spend a lot of time guessing – it’snot very efficient or effective. Just from $13/Page Order Essay