Question 2 Define a force field (in Newtons) by T 2y 2x F(x, y) = x + 1+x²y² 1 + x²y² where coordinates x and y are in meters. (a) Evaluate F(1, 0), F(0, -1), F(1, 1), F(-1, -1) and sketch them on the xy-plane. (8 marks) (b) (c) (d) Show that the force field is conservative and hence evaluate a scalar potential function E(x,y) such that F = VE. (9 marks) A path C is parameterized by the function below. Calculate the work done by F(x, y) along the path C. [t- r(t)= = 0 <t≤1. 1 (5 marks) With reference to the vector field, explain why the work done in part (c) is positive (or negative). (5 marks)

. A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F. (a) If the temperature of the turkey is 150°F after half an hour, what is the temperature after 45 minutes? (b) When will the turkey have cooled to 100°F?

The table gives estimates of the world population, in millions,(b) 1750 to 2000. (a) Use the exponential model and the population figures for1750 and 1800 to predict the world population in 1900and 1950. Compare with the actual figures. (b) Use the exponential model and the population figures for1850 and 1900 to predict the world population in 1950.Compare with the actual population. (c) Use the exponential model and the population figures for1900 and 1950 lo predict the world population in 2000.Compare with the actual population and try to explain the discrepancy.10.

3. A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide.What is the length of the longest pipe that can be carried horizontally around the corner? (Refer the diagram exercise 74 on page 269.)

Write the composite function in the form f(g(x)). [Identify the inner function u =g(x) and the outer function y = f(u).] (Use non-identity functions for f(u) and g(x).) (f(u), g(x))= y=\cos (\sin (x)) \text { Find the derivative } \frac{d y}{d x}

Suppose \sin (u)=\frac{3}{5} and cos (u) is negative. Then \cos (u)= \tan (u)= \sin (-u)= \cos (-u)= \tan (-u)= \sin (u+\pi)= \cos (u+\pi)= \tan (u+x)= \sin \left(u+\frac{\lambda}{2}\right)= \cos \left(u+\frac{\pi}{2}\right)= \tan \left(u+\frac{\pi}{2}\right)=

The graph of a function f is shown. Does f satisfy the hypotheses of the Mean Value Theorem on the interval [0, 5]? If so, find a value c that satisfies the conclusion of the Mean Value Theorem on that interval. (If an answer does not exist,enter DNE.) a) Yes, because f is continuous on the open interval (0, 5) and differentiable on the closed interval [0,5]. b) Yes, because f is continuous on the closed interval [0, 5] and differentiable on the open interval (0, 5). c) Yes, because f is increasing on closed interval [0, 5]. d) No, because f does not have a minimum nor a maximum on the closed interval [0, 5]. e) No, because f is not differentiable on the open interval (0, 5). f) No, because f is not continuous on the open interval (0, 5).

11-14 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. \text { 11. } f(x)=\left(x+2 x^{3}\right)^{4}, a=-1

12. Suppose that f(x, y, z. w) = 0 and g(x, y, z, w) = 0 determine z and w as differentiable functions of the independent variables x and y, and suppose that \frac{\partial f}{\partial z} \frac{\partial g}{\partial w}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial z} \neq 0 Show that \left(\frac{\partial z}{d x}\right)_{y}=-\frac{\frac{\partial f}{\partial x} \frac{\partial g}{\partial v}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial x}}{\frac{\partial f}{\partial z} \frac{\partial g}{\partial h}-\frac{\partial f}{\partial h v} \frac{\partial g}{d z}} and \left(\frac{\partial w}{\partial y}\right)_{\bar{x}}=-\frac{\frac{\partial f}{d z} \frac{\partial g}{\partial y}-\frac{\partial f}{\partial y} \frac{\partial g}{d z}}{\frac{\partial f}{\partial z} \frac{\partial g}{\partial w}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial z_{z}}} .

Consider the pictures below. Click on the pictures to see them more clearly. Each angle 0 is an integer when measured in radians. Give the radian measure of the angle.

Starting with the graph of y = e^x, write an equation of the graph that results from the following changes. (a) shifting 8 units downward (b) shifting 9 units to the right (c) reflecting about the x-axis (d) reflecting about the y-axis (e) reflecting about the x-axis and then about the y-axis