Find and plot the cubic spline S satisfying S(0) = 1, S(1) = 3, S(2) = 3, S(3) = 4, S(4) = 2 and with S' (0) = 0 and S'(4) = 1.

8) Suppose you are at the point (2, 1, 82) on the hill given by f(x,y) = 100-4x²-2y (where the z axis gives elevation, the y axis points north, the x axis east).Correct notation is required, units are not (13 points) (a) If you hike from the given point in the direction of = (3,4) are you going uphill or down? At what rate? (b) From the given point, in what direction is the steepest climb? (c) Using the graph of f(x,y) with the point (2,1,82) shown, explain and illustrate why your answers above are reasonable (or not).

An environmental engineer wants to evaluate three different methods for disposing of nonhazardous chemical waste: land application, fluidized-bed incineration, and private disposal contract. Use the estimates below to help her determine which has the least cost at /= 15.00% per year on the basis of an annual worth evaluation. First Cost AOC per Year Salvage Value Life Land $-145000 $-86000 $23000 4 years Incineration $-760000 $-70000 $260000 6 years Contract O $-132000 O 2 years

Help cubic function whose graph has horizontal tangents at (-2,6) and -- +5)=(OH • A 2 (2₁0). { y = ax ²³ + bx²² textd ах {Y= 1 X Y D(-2,6) (5) 6 f(2)=3ax²²³² +2bx+c =3a (2)² + 2b (2) tr 12a + 4b +c 3 Y = ax²³ + bxc²+cx + d 3 Y = ax² +bx² +ex td 3 6= a (-2)²³ + b (-2)² + (-2) + d (0= a (2) ²³+ b(2)³² + c(2) + d 0= 8a + 4b + 2 c + d 6=-8a + 4b - 2c + d 3 2 Y' (₂c) = ax²³ + bxc² +cx +d (ƒ(-2)=3ax²³² +2bx te 2 = a (3x²)+b(2x) + c(1) "3a (-2)² + 2b (-2) + C 2 = 3ax² +2bx te 8b+2d=6 8(0) + 2d=6 2d=6 622 d= 2 d = 3 YEW 1 +6=0 98a + 4b +2c tα = 0 X Y →→ (2.0) 8a-24a +3=0 -16 α = -3 -3 a=-16 3 а a = 16 -8a+4b-2x+d+86 +46+2c+d=6) 26=0 Jr) (³€+ N)=(3rk 4b+d+4b+d=6²² +²b = 2 b=2=6² b=0=₁ 8a +4(0) + 2 (-12a) +3=0 (2) 12a + 2b +c f(-2) s - 112a + 4b + c = (12a + 2b + c) = 0 12a + 4b+R-12a-2bx=0 2x604b =26=0m ³7 37 7₁ (10) 12a + 2b + c =0 -12a + 2(0)+c=0 12a+c=0 1)C=-12 au c=²=12a (²3² + 1) = (2x (c==12 (²76) 36 C = 76 -A= q 2 Y=ar² thre² textd 3 3 2 = nove +60₂e²+ ( ahe +3 2 3 3 = 16x² - 4x +3 льк