Write full, clear solutions to the problems below. It is important that the logic of how you solved these problems is clear. Although the final answer is important, being able to convey you understand the underlying concepts is more important. Be sure to write your name and recitation section on your homework, and be

sure to staple all loose pages.

1. (3 points) The energy, E, of a body of mass m moving with speed v is given by

????1???? E=mc ????1v2/c21.

2

Assume m > 0 and that v is nonnegative and less than the speed of light, c, which is a constant. Find E. What is the sign of E?

v v

2. (5 points) Find and graph the domain of the function f (x, y) = 6x + 3y 6.

1. (2 points) Suppose f is a differentiable function whose cross-section for y = 5 is f(x, 5) = 3x x2. Find fx(1, 5).
2. 2y
3. Letf(x,y)= x.
4. (a) (1 point) Show that the level curve f(x,y) = c is given by the curve y = log2(cx). (assume c > 0) (b) (2 points) Using the property log2(ab) = log2(a)+log2(b), show the level curves for c = 2 and c = 8
5. are y = 1+log2(x) and y = 3+log2(x), respectively. Sketch each of these curves on the same graph.
6. (4 points) Recall that a Cobb-Douglas production function has the form
7. P = cLK with c,, > 0.
8. Economists talk about
9. increasing returns to scale if doubling L and K more than doubles P ,
10. constant returns to scale if doubling L and K exactly doubles P ,
11. decreasing returns to scale if doubling L and K less than doubles P .

What conditions on the sum + lead to 1. constant returns to scale?

2. decreasing returns to scale?

Fully justify your answers mathematically.

6. (6 points) Find fx, fy, and fz for the function f(x,y,z)=zxy

(x>0, y>0, z>0).

(this is z raised to the xy power)

7. (4 points) The demand for a company's product depends on the price p it charges for the product and

on the price q charged for the product by a competing producer. It is f(p, q) = a bpq,

where a, b, and are positive constants with < 1.

Find f and f , and give a real-world interpretation of their signs. p q

8. (5 points) The Body Mass Index (BMI) of a person is defined by the equation BMI=f(m,h)= m

h2

where m is the person's mass (in kilograms) and h is the person's height (in meters), i.e. BMI is a function of the variables m and h. A rough guide is that a person is underweight if the BMI is less than 18.5; optimal if the BMI lies between 18.5 and 25; overweight if the BMI between 25 and 30; and obese if the BMI exceeds 30.

1. (a)If someone weight 64 kg and is 168 cm tall, what is their BMI?
2. (b)Sketch some level curves (contours) of BMI = f (m, h) for BMI = 10, 18.5, 25, 30, 38.5. Then shade the region that corresponds to the optimal BMI. Does Someone who weighs 62 kg and is 152 cm tall fall into this category?
3. (c)Draw the level curve (contour) of the BMI function corresponding to someone who is 200 cm tall and weighs 80 kg. Find the weights and heights of two other people with this same level curve (i.e. with the same BMI).
4. (d)Attempt to draw the surface plot corresponding to this function.