Calculus for Business

1. Find the following limits: a) 4 lim 6 1 5 x 4 x ?? x ?? ?? ?? b) 2 4 lim 2 8 x 4 x x ?? x ?? ?? ?? 2. Determine which statement best describes the continuity of the function: 2 3 2 ( ) 2 5 2 1 7 1 x if x f x x if x x if x ?? ?? ???? ?? ?? ?? ?? ?? ?? ???? ?? ?? a) f(x) is Continuous for All Real values of x b) f(x) is Continuous for All Reals except x = -2 c) f(x) is Continuous for All Reals except x = 1 d) f(x) is Continuous for All Reals except x = -2 and x = 1 3. Find the derivative of: f (x) ?? 7 ?? x ?? 4x2 ??5x4 4. Find the derivative of: 3 2 4 y x x ?? ?? 5. Find the equation of the line tangent to 3 2 2 2 1 2 y ?? x ?? x ?? at x ?? 6. Use Marginal analysis to predict the estimated profits from the 221st unit sold if C(x) ??1200??5x and R(x) ?? ??.04x2 ?? 32x ?? 200. 7. Use Implicit Differentiation to find dy/dx for: y3 ??5x ?? 6 ?? 3x2 y4 8. Find derivatives of the following: a) 2 ( ) 3 2 x h x x ?? ?? ?? b) ?? ??2 3 5 6 y ?? x ?? c) F(x) ?? ln??8x ?? 2x4 ?? d) ( ) 4 x h x ?? e 9. Find the derivative of: ( ) 3 x2 h x x e ?? ?? 10. Find the Absolute Maximum and Absolute Minimum of f (x) ?? ??x3 ?? 4x2 ?? 3 on [??1,5] 11. Find the derivative of: h(x) ?? ln??4x2 ?? 2x3 ?? 12. How long will it take an investment of $45000 to become $100000 if it earns 4.65% compounded continuously? 13. How much does it take invested today to grow to $30000 at 4.8% compounded continuously in 8 years? 14. Find the Elasticity function for the Demand function, D(x) ?? 800??4x , and then determine the elasticity at x = 75. ( ) ( ) ( ) x D x E x D x ?? ?? ?? ?? 15. Take first and second derivatives of f (x) ?? 54x2 ?? x4 , and then build a sign chart to determine Intervals of Concavity 16. Find the specific function f(x) satisfying the following conditions: f ??(x) ?? 6x2 ?? x ??2 and f (4) ?? 5 17. Evaluate the Integral: ?? ?? 3 3 1 4x 4x 3 dx ?? ?? ?? ?? 18. Find the value of the each Definite Integral: a) ?? ??1 3 2 3 2 x x 3 dx ?? ?? ?? b) 1 2 3 1 2 x x e dx ?? ?? ?? c) 10 2 2 2 4 x dx x ?? ?? 19. Use Integration by Parts: ( 2) 4x ?? x ?? e dx 20. Find the Area bounded by y ?? x ??5 and y ?? x2 ??6x ??11 21. Tim invests money in a 403b account according to the function ( ) 15000 .02t f t ?? e . If the account earns continuous gains at 5.5%, and the contributions are made for 25 years, what will be the value of the account? Use: 0 ( ) T kT kt FV e R t e dt ?? ?? ?? and round to the nearest $1. 22. An investor is looking at a business opportunity, which will pay him $100000 each of the next 8 years. Assuming continuous growth at 6%, what is the Present Value of this income stream? Use: 0 ( ) T kt PV R t e dt ?? ?? ?? 23. Find ?? ?? 2 2 ( 3,2) ( , ) 4 2 xy f if f x y x y ?? ?? ?? 24. Find Critical Points and then whether they are a Relative Max, Relative Min, or Saddle Point. f (x, y) ?? x2 ?? y2 ?? xy ?? x3 25. The function f (x, y) ?? 2x4 ?? y2 ??12xy has derivatives: 8 3 12 , 2 12 x y f ?? x ?? y f ?? y ?? x . Find the critical points and determine whether each is a Relative Max, Relative Min, or Saddle Point. 26. Find the Partial Derivative xy f for f (x, y) 1 ln x y ?? 27. Find ALL 5 first and second order Partial Derivatives of: a) P(x, y) ?? 2x4 ?? y5 ?? 4xy6 b) ( , ) 2 6 x y f x y y x ?? ?? 28. Use the Method of Lagrange Multipliers to find the Maximum and Minimum of f (x, y) ?? 2xy subject to x2 ?? y2 ??18