Student number Name [SURNAME(S), Givenname(s)]
MATH 100, Section 110 (CSP)
Weeks 10 and 11: Marked Homework Assignment
Due: Thu 2010 Nov 25 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. For the function x 2 + 3 f ( x ) = ,
2 x − 2 determine all of the following if they are present: ( i ) critical numbers, local maxima and mimima, intervals where f is increasing or decreasing; ( ii ) inflection points and intervals where the graph is concave upward or downward; ( iii ) asymptotes (horizontal, vertical, slant). Sketch the graph of y = f ( x ), giving the ( x, y ) coordinates for all the points of interest above.
2. Let f ( x ) = x
√
3 − x . (a) Find the domain of f . (b) Determine the x -coordinates of the local maxima and minima (if any) and intervals where f ( x ) is increasing or decreasing. (c) Determine intervals where f ( x ) is concave upwards or downwards, and the x -coordinates of inflection points (if any). (d) There is a point at which the tangent line to the curve y = f ( x ) is vertical. Find this point. (e) The graph of y = f ( x ) has no asymptotes. However, there is a real number a for which lim x →−∞ f ( x )
| x | a
= − 1 .
Find the value of a . (f) Sketch the graph of y = f ( x ), showing the features given in items (a) to (d) above and giving the ( x, y ) coordinates for all points occurring above and also all x -intercepts.
3. A hiker starting at a point P on a straight east-west road wants to reach a forest cabin that is 2 km due north of a point Q on the road which is 3 km from P . Assume the hiker can walk 8 km/h along the road, but only 3 km/h through the forest. How far down the road should the hiker walk before setting off through the forest straight for the cabin, in order to reach the cabin as quickly as possible?
4. Find the length of the shortest ladder that can extend from a vertical wall, over a fence
8 ft high located 1 ft away from the wall, to a point on the ground outside the fence.
5. A drinking cup in the form of a right circular cone is made from a circular disk of paper of radius R . A sector of angle 2 πα , 0 < α < 1, is cut out of the disk and the remaining part of the disk is bent up so that the two straight edges join and a cone is formed. Find the value of α which gives the cup of maximum volume. [ Hint: First find the circumference of the base of the cone.]