# 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?