Calculus

Instructions: Do all of your work on this paper. Scan the completed problems and create one PDF of your work. Your PDF needs to be at least the same number of pages as the original docu- ment. Upload the PDF to this assignment in GRADESCOPE by the due date. (The Gradescope assignment can be accessed through Canvas.) (1) Let f(x) = cos²(x) sin(x²). Determine f'(x).

3. Find the volume of the region bounded above by the cylinder z = 4 - y² and below by the paraboloid z = 2x² + y².

Turn in these problems: 1. Determine whether each sequence below converges or diverges. If the sequence converges, find its limit. Carefully show your work!

2. Determine whether each series below is convergent or divergent. If the series is convergent, find its sum. Carefully show your work, and justify any tests that you use.

3. Suppose f is a continuous, positive, decreasing function for a 21 and ak = f(k) for k 2 1. By drawing a picture, rank the following three quantities in increasing order:

3) Find the slope of the tangent line to the graph at two different points on the graph. The points chosen are up to you but they should be "points of interest"

12. The initial point of vector MN-[2, 4, -7] is M(-5, 0, 3). Determine the coordinates of the terminal point, N.

(5) The number of deer in a National Park is modeled by P(t), where time t is given in years. 1070€0.23t 0.9 + €0.23t P(t) Use the Intermediate Value Theorem to show that the model predicts there will be exactly 1000 deer at some time t, where 10 < t < 15. You do not need to find the exact value of t that gives 1000 deer.

3. (§5.2, 5pts each) Each graph below represents the velocity function of an object moving in a straight line. For each graph, determine the following: • What is the object's maximum velocity? What is the object's maximum displacement? What is the object's total displacement? Remember to include the proper units for your final answers.

2. A ladder 5 m long is leaning against a vertical wall. If a person pulls the bottom of the ladder away from the wall at a rate of 0.5 m/s, how fast is the top sliding down the wall when the bottom of the ladder is 3 m from the wall?