Business Calculus II Final Review: Integrals, Definite Integrals, Areas, Differential Equa, Exams of Mathematics

Download Business Calculus II Final Review: Integrals, Definite Integrals, Areas, Differential Equa and more Exams Mathematics in PDF only on Docsity! Business Calculus II Final Review Sheet Remember our final is on TUESDAY MARCH 15 at 3.15pm. 1. Calculate the following indefinite integrals. (a) ∫ (2x + 1 + x−1)dx. (b) ∫ 3x2+2 x3+2xdx. (c) ∫ 3x2+2 (x3+2x)3 dx. (d) ∫ (3x2 + 2)ex 3+2xdx. (e) ∫ 3x2+2 x3+2x ln(x 3 + 2x)dx. 2. Calculate the following definite integrals. (a) ∫ 1 0 x 3dx. (b) ∫ 1 0 x2 x3+1dx. (c) ∫ 1 0 xe x2dx. 3. Calculate the areas of the following regions. Do not count areas below the x-axis as negative! (a) The region enclosed by y = x and y = −x4. (b) The region between y = x, y = x3, x = −1 and x = 1. (c) The region enclose by y = ex, y = 2 and the y-axis. 4. Sales of ice cream are continuously rising at a rate of 10 percent per month. My ice cream company currently sells 1000 quarts each month. Write down a differential equation describing the change in sales and then solve it to predict my monthly sales in 6 months time. 5. Find the equation for the tangent plane to the following surfaces at the point (1, 2, 3). (a) z = x2 + 3xy − 2y − 3. (b) z = 7x− 3y + 2. 6. Find all the critical points of the following function, and decide if they are local minima, local maxima or neither. (a) f(x, y) = x2y − 2x2 − 4y2. (b) f(x, y) = x2 + y − ey. (c) f(x, y) = x2 + y2 + 2xy . (d) f(x, y) = ex 2+y2 . 1