(1) Find the derivative function for each of the following: (a) f(x)=x2x-1 (b) f(x)=sin(3-x) (2) Evaluate each integral. 1 (a) dx (b) [sece tane de (c) f(x-1)x dx 3x5 x+5x+4 2 x+4x+3x-4 (e) lim (3) Find each limit (Hint: use simplifying/rewriting techniques, or degree shortcuts, or when appropriate you can just use L'Hopital's Rule instead) (a) lim x + 5 x-5 2x -3 3x - 4x+1 2x +1 (b) lim x2 x +6x-4 2 x 5x +6x -10 (4) For the function f(x) = 2x - 6x find the following: (a) All x values for which f(x) is increasing. (b) All x values at which f(x) has a relative minimum. (c) All x values at which f(x) has a relative maximum. (d) All x values for which f(x) is concave down. (e) All points of inflection on f(x). -2 4- 3 2+ (1.4) (c) h(x) = (c) lim (5) (TI-84 required) A particle moves along an x-axis with position at time t given by x(t) = t sin 4t for the time interval 0t1.5. Its position at time t = 0 is x(0)=0. - N- (a) Graph the velocity function on the given interval. (copy a rough sketch) (b) Find all times t on this interval when the particle is moving left. (Hint: when V is neg.) (c) Find all times t when the particle changes directions. (Hint: When V(t) changes from positive to negative or vice versa, the particle will change direction) (d) Find the total distance traveled for the particle during the interval .5 t1. 2 X (6) Use the sketch of f below along with the fact that g(x) = f(t) dt to answer the questions $5(1) d which follow. (Hint: BEFORE parts c through e, you will need to differentiate both sides) (2.1) 4x+1 x-2 3 2 x - 4 xx2+x-4x (d) lim (4,-1) X- (2x-1) (a) Find g(4) (b) Find g(-2) (c) Find the absolute maximum value of g(x) on the interval [-2, 4]. (d) Find the intervals on which g(x) is concave down. (e) Find all values of x at which g(x) has a point of inflection. SX 10%