Exam 1 Feb. 26, 2009 SHOW ALL WORKMath 28 Calculus III Either circle your answers or place on answer line.[14] 1.) Use the chain rule to calculate D(f ◦ g)(s, t) where f : R2→ R3,f(x, y) = (x, y, exy) and g : R2→ R2, g(s, t) = (t2, sin(st)).Answer: D(f ◦ g)(s, t) =[14] 2a.) Suppose f(x, y) = exy. Approximate f(1.9, 0.1) by finding a best linear approx-imation to f at an appropriate x = a.Answer 2a: f(1.9, 0.1) ∼[6] 2b.) D(3,4)f(10, 2 ) = where f (x, y) = exy.[5] 3a.) proj(1,2)(8, 6) =[4] 3b.) Suppose that a force F = (8, 6) is acting on an object moving parallel to thevector (1, 2). Decompose the vector (8, 6) into a sum of vectors F1and F2where F1points along the direction of motion and F2is perpendicular to the direction of motion.Answer 3b: F1=, F2=[1] 3c.) Verify that F = F1+ F2[4] 3d.) Use the dot produc t to verify that F1and F2are perpendicular to each other.Explain how the dot product can be used to verify that two vectors are perpendicular.[12] 4.) Find the following limit if it exists. If it doesn’t exist, state why you know itdoesn’t exist.lim(x,y)→(0,0)2x2− y2x2+ y2[5] 5.) State the limit definition of differentiable:f : Rn→ R is differentiable at x = a if[12] 6a .) Let f : R2→ R, f (x, y) = x2+ 4y2. Draw several level curves of f (makesure to indicate the height c of each curve). Draw vectors in the direction of the gradientof f at (√12, − 1) and at (0, 2). The length of your vectors should denote their relativemagnitudes.[2] 6b.) Identify the quadric surface in 6a:[12] 7.) State the equation for the li ne of intersection of the planes 2x − y + 3z = 10 and4x + 5y − 10z = 20Answer8.) Circle T for True and F for False:[3] a.) Supp ose f : Rn→ R. If f is differentiable, then∂f∂xi(a) T Fexists and is continuous for i = 1, ..., n.[3] b.) Supp ose f : Rn→ R. If∂f∂xi(a) exists and is continuous T Ffor i = 1, ..., n, then f is differentiable at a.[3] c.) Supp ose f : Rn→ R. If Dv(f)(a) exists for all v, T Fthen f is differentiable at a.[3] d.) If f is continuous, then f is differentiable. T