2. (10 points) A portion of the graph y = f(x) is shown in the image below, and the region bounded by this graph and the r-axis is shaded. 2 Figure 1: Graph of y = f(x) over [0, 4). If we are told that / f(x) de = 16/3, / f(x) dx = 37/12 and /"f(x) dx= / f(x) dx evaluate the definite integral [ f(x) da =3. (10 points) Two graphs are shown below (along with the shaded regions bounded by them and the r-axis). A) (5 points) Calculate the deifnite integral of the graphed function without usig the Fundamental Theorem(s) of Calculus. Be sure to explain how you calculated this number.) Figure 2: Graph of f(r) = 3 + v10r - x2 - 21. 3+ V10r - 12 - 21 dr = B) (5 points) Calculate the deifnite integral of the graphed function without usig the Fundamental Theorem(s) of Calculus. (Be sure to explain how you calculated this number.) Figure 3: Graph of g(x) = |4 + 4x].4. (10 points) A graph of a function y = f(t) is shown below. Use this graph to answer the following questions about the function F(x) = / f(t) at. y = f(1) -3 -1 Figure 4: Graph of y = f(t) A) (3 points) Compute the values of F(1) and F'(1). B) (7 points) Find the maximum and minimum values of F(x) on the interval [-4, 2].5. (10 points) The functions F(x) and G(x) are given by F(x) = cartand at and G(z) = earctant it Note: Like the function e , the integrands used to define both F(x) and G(x) are not possible to anti-differentiate in terms of our old, familiar functions, so don't waste your precious time trying. A) (2 points) We can express the function G(x) as a composition of F(r) with another function; that is, we can write G(z) = F(h(r) ). Identify the "inside function" h(I). B) (4 points) Find F'(x) and G'(z). C) (6 points) Check that L'Hopital's rule applies to the limits lim F(x) and lim G(I) -+0 and use this rule to evaluate these limits.6. (40 points) Complete the following cross-word puzzle with English words (this means writing numbers out as words). showing your work for clues 4, 7, 8, 12, 13, and 14 Across 1. (5 points) A function /(x) with derivative f'(x) = ex (5 - x)3 has a local at x = 5. 4. (5 points) (In7 - 73 dx) + 1 6. (2 points) An anti-derivative for the exponential, ez, is the B. (5 points) / vitzdat / aldr 9. (5 points) (1+x3)-1 dx = (x) + C. 11. (0 points) When computing an indefinite integral, don't forget the _ 12. (2 points) One half of Clue 8 minus the value 15 . F(w/4) where F(x) is the function used in Clue 9. 14. (points) The score you will earn on your final; a.ka. the minimum value of /(x) = 89ex* + 11. Down 2. (2 points) If a particle is traveling along a horizontal access with velocity s'(t) = v() = 30t + 5 ft/see then its is 30 ft/see?. 3. (2 points) The definite integral v2 + cosi dix equals the area under the graph v2 + cosr over [0, 2x]. 5. (2 points) The technique of integration that "undoes the Chain Rule" (and was also used in the first integral of Clue 8). 7. (3 points) The area under the curve of y = see I over [0, x/4]- 10. (5 points) If f'(x) > 0 over an interval, then the y-values of f(x) " 13. (2 points) The value of & that satisfies / feldt = 2. fit de - 817. (10 points) This problem consists of two parts. A) (7 points) Evaluate the following indefinite integral 5 In(r + 1 dr. 1+23 + c+1 B) (3 points) Based on your work in part A), find a function y(r) that satisfies the differential equation y' (x ) = 1 In(a + 1) +1 and initial condition y(0) = 5