This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Review materials for the Final Exam, MTH 142, Spring 2008 The Final Exam is Monday, May 12 , 8  10:20 AM in LDB 400. Calculators will not be allowed on the Final Exam. As always, you will be graded on your work . Problems from Exams 2 and 3 1. Two boats meet in the middle of the ocean. At noon, one boat heads north at 30 miles per hour. The other boat stays at the meeting point for an extra two hours, then, at 2 PM, heads west at 20 miles per hour. (a) At 4 PM, what is the distance between the boats? (b) At 4 PM, how rapidly is the distance between the boats changing? (Show your work/explain your reasoning.) 2. The graph of y = f ( x ) is given below. Using that graph, draw in the space provided below it, the graph of the first derivative. Label any critical points. 3. Suppose f ( x ) = x 4 4 x 3 + 4 x 2 + 18 . (a) Use the first derivative to find the critical points of the graph of y = f ( x ) and determine which are local minima and which are local maximum. (b) Use the second derivative to find inflection points of the graph of y = f ( x ) . (c) Determine where the graph of y = f ( x ) is concave up and concave down. (d) On the additional paper provided, carefully draw a graph of the curve y = f ( x ) . Label all points found above. 4. Find dy dx . (a) sin y + x 2 y = πe 2 . (b) y 3 + xy 2 = cos x. (c) y = ln( x 3 + 5 x ) (d) y = ln(sec x ) 5. Compute dy dx . Put your answer in terms of x if possible. (a) ln y = x ln x (b) sin y = x 6. A girl flies a kite at a height of 50 meters. The wind carries her kite horizontally away from her at a rate of 10 meters per second. How fast must she let out the string when the kite is 130 meters away from her? 7. (a) i. Find the equation for the line through (0 , 1) with slope 1 . ii. Find the equation for the line through (4 , 2) with slope 1 4 . (b) Linearize each of the functions f ( x ), below, at the point ( a, f ( a )) . i. f ( x ) = e x , a = 0 . ii. f ( x ) = √ x , a = 4 . (c) Use your work in part (b), above, to estimate the values of: i. e . 01 . ii. √ 3 . 9 . 8. Suppose x is a positive real number. (a) Find a formula for sec(tan 1 ( x )) . (b) Find a relationship between tan 1 ( x ) and cot 1 ( x ) . 9. In this problem you will model changes in a certain spherical hailstone bobbing up and down in a thunderhead. Please make sure your answers have correct units. (a) Suppose the hailstone has radius 5 mm. As the hailstone rises in the cloud, another layer of ice is added to the surface of the hailstone, increasing its radius by 0.2 mm. Use differentials to estimate the change in volume of the hailstone due to this increase....
View
Full Document
 Spring '08
 STAFF
 Calculus, Derivative, Mathematical analysis

Click to edit the document details