2. The approximate rate of change is p'(t) = 43.14t - 143.5, where t represents the number of years since 2000, and p represents the number of patents (in thousands). In 2008, 828,328 patent applications were received. a. Find the function that gives the total number (in thousands) of patent applications received in China in year t. b. According to your function from part (a), how many patent applications were received in 2015? Round to the nearest whole number. 3. Find the cost function, C(x), if the marginal cost function is C"(x) =- 8 - and the fixed cost (when 2x+1 no units are produced) is $18.1. Find each of the following integrals. Some (but not all!) will require integration by substitution. Make sure your workotation is written out correctly as well. a. | ( 2 71 + 7 2 - VZ ) dz b. 7x(4x2 - 1) dx C. 4xsex dx d. 2 In x dx x e. V8 - x dx F. (3x2 - 5)4(2x) dx