|x \ y||-1||0||1||2|
|x \ y||-3||-2||-1||0|
3. Let f(x,y) = 2y4 - 3y3 x2 + 2xy - x3 + 1.
Find fx , fy .
4. Suppose that your weight w, in pounds, is a function f(c,n) of the number c of calories you consume daily and the number n of minutes you exercise daily. Using units, interpret in everyday terms the statements:
(a) w(2100,20) = 120 (b) wc (2100,20) = 0.03 (c) wn (2100,20) = -0.025 .
Estimate w(2300,20) and w(2100,25).
5. Find the global maximum of the function f(x,y) = y2 + x2 in the region R: 0<=x<=1, 0<=y<=2. Explain your reasoning.
6. Find local minima, maxima and saddle points for the function f(x,y) = 2y3 + 3 x2 - 6xy.
7. Verify that the function f(x,y) = y4 + x3 has a critical point at (0,0). Is it a local maximum, minimum or a saddle point ? Explain your answer.
8. A contour diagram of a function f(x,y) is given below. The function has critical points at (0,0) and (1,1). Classify each of them as a local minimum, maximum or a saddle point. Explain your answer.
9. A cruise missile has a remote
guidance device which is sensitive to both temperature and humidity. Army
engineers have worked out a formula to show the range at which the missile
can be controlled, f(t,h),
miles, as a function of the temperature
in degrees Fahrenheit, and
percent humidity h:
f(t,h) = 12,000 - t2 - 2 ht - 2 h2 + 200 t + 260h.
What are the optimal atmospheric conditions for controlling the missile ?
10. Consider data points (1,1),
(2,2.5), (3,2.5), (4,5). Suppose you want
to find a least squares line y = mx
+ b for these data points without using
a calculator program. What function f(m,b)
would you have to minimize ? Write the function f(m,b)
but do not perform actual calculations.