Fundamental Theorem of Calculus

1. Evaluate by hand. Be sure to show all work. (2 points each)

a)​ ​∫ dx8x √2−x2

b)​ ​∫ e dx x2 + x + 3 2. Evaluate each of the definite integrals by hand using the Fundamental Theorem of Calculus. Be sure to show all work. (2 points each)

a)​ te dt∫ 1

0 2 t


b) dx∫ 16


1 √x

3. (x, ) − xy f y = x4 − y4 + 4 a) Find all the critical points for this function. (2 points) b) What point is a local maximum value of f(x,y)? Prove your answer using the second derivative test. (2 points) c) Does the function have a saddle point? If so, what is the function’s value at this point? (1 point) 4. Consider demand and supply equations 300 2q, p .04q 0. p = − = 2 + 1 a) Find equilibrium price and quantity. Assume that q must be produced in whole units. You may use technology instead of doing this by hand. (1 point) b) Calculate Consumer Surplus. (2 points) 5. Suppose a particle travels according to velocity . ​Units of v(t) are meters/second. Find the(t) v = t

1 total distance the particle travels between t = 1 seconds and t = 6 seconds. You do not need to simplify your work. (2 points) 6. (1 point each) True or False: a) If price is artificially raised from its equilibrium point then Consumer Surplus + Producer Surplus will increase. b) In general, .f f xy = yx c) If f(x,y) has a critical point at (a,b) and , then (a,b) is either a local maximum or af f 0 f xx yy − f xy yx > local minimum. d) The average value of between x = 2 and x = 6 is 52.(x) 3x f = 2 e) The curve on a contour diagram of f(x,y) shows a constant value for f(x,y) as x and y vary. f) For demand equations , goods represented by q​1​ and q​2500 p 5p , q 200 p p q1 = − 1 − 2 2 = − 4 1 − 2 are substitute goods.

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