4. (Integration) Calculate each indefinite or definite integral, if possible. 2 ) / 2 1 /3 + 1 do b) / cos 3 ( 0 ) sin ( 0 ) do c ) M eVI - da 4. 1309. (Fundamental Theorem of Calculus / Interpreting Graphs) A graph of a function f(x) is shown below. Define the function F (I ) = $ (s ) ds Answer the following true or false questions about F(x). You may provide a justification for each answer for partial credit. N f (I) true or false (a) The derivative of F at x = 1 is zero, F'(1) = 0. (b) F(x) is increasing at x = 1.5. (c) The derivative of F(x) at T = 3 is zero, F' (3) = 0. (d) F(x) is concave down at x = 4.5. (e) F(x) has an inflection point at x = 6. (f) F(x) evaluated at r = 10 is zero, F(10) = 0. 9. /10(Applied Optimization) You are building three identical pens adjacent to each other with a total area of 450 m, as shown in the following figure. What dimensions should you use to minimize the amount of fencing? y -C- 8.(Limits) Evaluate each limit using any method of your choice. a) lim x + sin c 1-+0 x + cos x b ) lim V9x2 + 1 x- -00 2x+ 1 C) tan '(f(x) ) lim where f(0) = 0 and f(x) is differentiable. I-+ f (I ) 1. 130\f3. (Implicit Differentiation) For the following functions of r (defined explicitly or implicitly), find the derivative with respect to r. a) xy - cos(ry) = 1 b) 3x2 + 4xy' = ry c) y = (cos x) 3. 13010. (Final challenge problem) The graphs of a function f(r) and g(I) = In(f (x) ) are shown below. Calculate exactly (meaning no decimals) the value of f' (4). 20 f(x) g(x) = In [f (x) ] 3 2.5 15 2 10 1.5 51 0.5 2 3 4 5 6 2 3 4 5 6 10. /107. (Related Rates) A tank shaped like an inverted cone is leaking water, but you do not know at what rate. The tank has a height of 16 ft and a radius of 5ft. If the water level is decreasing at a rate of 0.25 ft / min when the depth of the water is 8 ft, determine the rate at which water is leaking out of the cone. You may use that the volume of a cone with height h and radius r is given by V = 3Trzh. 7. /10