{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math 100 Dec 01 Questions

Math 100 Dec 01 Questions - MATH 100/180 December 2001...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 100/180 December 2001 Question 1 [54 marks, 3 marks each] a) b) C) d) g) h) j) k) 1) x2 ~x—2 Evaluate lim 2 X—>2x +x——6 Determine an equation of the tangent line at (1,2) to the curve y = x2 — 4x + 5 . Calculate the derivative of (l —— x2)sin‘l x 2 — Calculate and simplify the derivative of 32—3. x — 2x + 2 If f(0) = 5 and f'(0) =% find the derivative of g(x) = 1+|f(x)|2 at x = 0. Acosx if x20 Suppose the function f is defined as f (x) = _ for constants A and B. For what 1— Bx If x < 0 values of A and B is f differentiable for all x. Find the third derivative of f (x) = sin ax where a is a constant. Find the tangent line at (0,0) to the curve tan‘1 (x + y) + x2 — 2 = 0 . A bead is moving on a straight wire and its position is given by x(t) = 4t3 —t . Find the positive time t at which its instantaneous speed is equal to its average speed over the time interval [0,1]. A bead is moving on a straight wire and its position is given by x(t) = t4 — t2 . Find all times I when the acceleration is zero. If f (3) = 5 and f ' (3) = 2 use a linear approximation to estimate the value f (2.8) . A spherical balloon is being filled with air at a constant rate of 6 cubic centimeters per second. When its radius is 10 cm, how fast is the radius increasing. Ignore any compressibility effects of the air. A bacteria colony is established and then grows exponentially. If there are 10,000 bacteria after 1 day and 20,000 after 2 days, how many will there be after 5 days? Newton’s Law of Cooling states that temperature T(t) of an object changes with time t at a rate proportional to the difference between T(t) and the ambient temperature To (the temperature of the surroundings). Express this law as a differential equation for Tusing a positive constant k. Do not attempt to solve this equation. Indicate on the g raph below where the estimates x1 and x; to the roots of f(x) will be located if calculated using Newton’s Method starting with initial estimate (guess) x0 f(x) X0 X p) Determine the first three nonzero terms in the Taylor series based at x=0 for e)‘2 . q) Determine the Taylor series based at x=0 for (x + 1) sin x showing all terms upto and including the x3 term. 5 ex2 . c0 x — r) Evaluate 11m ———2—— x—>0 x Question 2 [10 marks] You are flying a kite. At a certain time, the kit is 30m high and 40m horizontally away from you and is moving horizontally away from you at a rate of 10 meters per minute. Assume the string lies on a straight line between you and the kite at all times. a) How fast are you letting out the string at that time. b) How fast is the angle between the string and the ground changing at that time. Question 3 [12 marks] A window is in the shape of a rectangle with the top edge replaced by a semicircle and has perimeter 10m. Find the dimensions of the rectangle that gives the window of greatest area. Question 4 [12 marks] A certain radioactive substance is known to have a half live of 120 years. Initially, 100 kg were placed in an underground storage facility in 1980. In 2000, another 100 kg were added in a single shipment and in 2020 another 100 kg will be added in a single shipment. How much of the radioactive substance will there be in the facility in 2030'? Question 5 [12 marks] A function f(x) is known to be defined and have continous first and second order derivatives for all x. It is also known to have the following properties: i) f(x) : —f(—x) for all x (ie the function is odd) ii) the graph of f has a slant asymptote y=x iii) the only value x>0 for which f '(x) = 0is x=1 iv) the only value x>0 for which f ” (x) = 0 is x=2 v) f(1)=_1,f(2)=1,[email protected]=o Sketch a graph of f(x) that satisfies all of the conditions above showing all root(s), local minina(s), local maxima(s), and inflection point(s). ' ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern