MAT 21A, practice problems for the final exam 1. Compute the limits: a) limx3 e1/x b) limx3 ln(x 3) c) limx3 ln(x2) x3 d) limx 8x5 7x3 +9 (3x2 1)(2x3 3) e) limx ex x3 f) limx0 arctan(x)x x3 2. Compute the derivatives of the following functions: a) f (x) = x ln x x b) f (x) = e3x 2 c) f (x) = (x 1)5 cos x d) f (x) = sin(ln x) e) f (x) = e3x+2 cos x+2 3. Find the minimal and maximal values of a function: a) f (x) = x2 ex on [0, 1] b) f (x) = 2x3 3x2 + 1 on [1, 2] c) f (x) = sin2 x on [0, ] d) x 1+x2 on [2, 2] 4. Find the equation of the tangent line to the graph of f (x) = ln x at x = 5. 5. For a given function: Find the domain Determine the equations of vertical and horizontal asymptotes Find the derivative and determine the intervals where the function is increasing/decreasing 1 Find the second derivative and determine the intervals where the function is concave up/down, find inflection points Draw the graph using all the information above a) f (x) = 2x3 3x2 + 1 b) f (x) = xex c) f (x) = ln(x2 + 1) d) f (x) = x1 x+1 6. Consider the function ( x + 1, if x < 1 f (x) = 2 x + ax + b, if x 1. a) For which values of the parameters it is continuous? b) For which values of the parameters it has a derivative at every point? 7. Consider the curve given by the equation x2/3 + y 2/3 = 1. Find y 0 using 3 implicit differentiation and sketch this curve. (Note: assume that x2/3 = x2 is defined for all x). 8. Consider the curve given by the equation y 2 = x3 x. Find y 0 using implicit differentiation and sketch this curve. 9. An open rectangular box with square base is to be made from 1 area unit of material. What dimensions will result in a box with the largest possible volume ? 10. A TV set costs 100. If its price is lowered by a%, the sales would increase by 2a%. Find the discount amount a which yields the maximal profit. 2 Scanned by CamScanner