MATH 203 Final Examination December 2019 Page 2 of 3 [5] 6. Consider the following piecewise-dened function: f() 4a:172 if 1731 a}: (721: if a:>l Find the values of a and b so that the function f (x) is differentiable at every :13. Sketch the graph of f (x) on the interval [3, 3]. [6] 7. (a) Calculate the differential df of the function f = tan(a:). Use it to find an approximation for the change of f in the vicinity of a: 2 7T / 4, i.e. Af = f(:z:) f(7T/4), in the form of differential df at a: = 77/4. (b) Use this differential to approximate tan(7T / 3) given that g = g + 132 . [7] 8. Let f(a:) = \\/1 +231:2 . (a) Find the slope m of the secant line joining the points (0, f(0)) and (2, f(2)). (b) Find all points :1: = c (if any) on the interval [0,2] such that f'(c) = m. [12] 9. (a) Verify that the point (2,0) belongs to the curve defined by the equation 3/ + :Lw/l + y2 = 31:2 2, and nd the equation of tangent line to the curve at this point. dl (b) The length l of a rectangle is increasing at the rate a = 10 CE while its width s dw w is decreasing at the rate E = 5 (3100. When the length is 20 cm and the s width is 8 cm, how fast is the area of the rectangle changing at that instant? Is it increasing or decreasing at that instant? Ll? xgmll [14] 10. (a) Find the absolute extrema of f(:1:) = on the interval [0, 3] . b A rectan le ABC D has sides arallel to the coordinate axes, oint A is at the g P p origin, point B is on the positive xaxis and point C is on the curve y 2 67I/2. Find the maximum possible area of the rectangle (do not approximate). \" 2 3 (c) Use l'Hopital's rule to evaluate the lim 9:40 1 cos(2:1:)