Questions in Section Calculus I: page 7

Given the parametric curve given by `{(x=1+2t),(y=e^t+e^-t):}, 0<=t<=2`.

Find the slope of the tangent line at the point `t=1`.

Draw a tangent line segment.

Also give the four-decimal approximation.

Determine the Taylor polynomial of order 4 for the function `g(x)=e^(x^2)cosx` about `x=0` and use it to find `lim_(x->0)(1-e^(x^2)cosx)/x^2`.

[Hint: The limits is of the type `0/0`, so you can use L'Hopital's rule to check your answer.]

  1. Using the Taylor polynomial of order 5 for `e^u` about `u=0`, write down the Taylor polynomial of order 5 (up to and including `x^5`) for the function `f(x)=(1-e^(-x/2))` about `x=0`.
  2. Using your answer to part a) of this question (or otherwise), write down the Taylor polynomial of order 4 for `g(x)=f(x)/x` about `x=0`.
  3. Plot `g(x)` and the Taylor polynomial from part b) on the same graph for `x in[-2,2]` and comment.

Test the convergence (i.e., find the limit, if it exists) of each of the following infinite seqences with n-th terms given by:

  1. `a_n=5/(2n^2)-(2/3)^n`
  2. `b_n=(n^2+3n-2)/(7-4n^2)`
  3. `c_n=(-1)^n(2-3n)/(5-2n)`

Derive the following identity: `(d(text(arccosh)(u)))/(dx)` = `1/(sqrt (u^2-1))` `(du)/(dx)`, `u>1`.

Using L'Hospital's rule, if appropriate, determine `lim_(x->0)((x+1)^2-1)/(x^2-sin(2x))`.

State and determine all necessary limits for sketching `y=(e^(-x))/(1-x^4)`.

A horse on a merry-go-round moves in such a way that its height (in metres) above the floor is `h(t)=0.5sin(((t-2)pi)/2)+1.5` where `t>=0` is time in seconds.

  1. Using the formula for `h(t)`, find its period. Sketch the graph of `h(t)` for `0<=t<=8`.
  2. Find the time at which the horse reaches its maximum height for the 50th time, showing the reasoning.
  3. Find `h'(t)` and evaluate `h'(0.5)` exactly. Is the horse moving up or down at `t = 0.5` and why?
  4. Find the smallest value of `t>=0` for which `h'(t)` is a maximum, showing the working.

Consider the crank-slide mechanism shown below. The crank or driving link `OA` revolves counterclockwise about the fixed point O at a constant angular velocity. The length `|OA|` is 1 dm (1 dm = 10 cm). The driven link `AP` of length 4 dm reacts so that the end P is constrained to move along a horizontal line `BC` (where B is directly above O). There is a barrier at D to prevent the end P moving to the left of B. The distance `|OB|` is 3 dm. At any given instant, let the angle of rotation of the crank be denoted by `theta` and the distance `BP` be denoted by `x`, with `x>=0`.

A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?

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