Please Note: Max mins must be proven with FDT or second derivative. If you draw a graph, for full marks, label everything you discovered, including asymptotes and draw arrows and show coordinates of points where applicable. Knowledge and Understanding (6 marks) 1. Consider the following polynomial function graph of f(x) (Use the letters as a guide and state your answers in terms of these letters if applicable.) a) State the interval(s) over which f'(x) is positive. b) State the interval(s) over which f"'(x) is negative. c) State the interval(s) where the product of f'(x) and f"'(x) is negative. d) State the minimum degree of the polynomial. Explain why. e) Which point(s) correspond to global minimums? f) Which point(s) correspond to absolute maximums? Name: Date: 2. Below is the graph of the first derivative of a function y = f(x). Sketch in the next derivative and determine the intervals where the function y = f(x) is concave up and where it is concave down. Make marks on the graph if necessary. [2 marks] Write your answer here: Explain the proper logic. 3. The following graph shows the first 3 derivatives of a function. Label the first, second, and third derivatives, and draw a sketch of the original function. 2 marks Name: Date: Communication (8 marks) 4. Sketch a graph of a rational function that satisfies the following conditions: (3 marks) f(0)=0, f(-3)=-5, f(3) =5 f(x) is undefined for x =12 y S'-3)=1(0)=1(3)=0 f'(x)0forx3 4 f0)=0 f"(x)0for-22 The line y = x is also an oblique asymptote. Label fully. 5. True/False: (5 marks). Explain the correct logic below each question. a) If the second derivative = 0 at x = a, then you will have an inflection point at x = a. b) If the first derivative = 0 at x = a, then you will have a max or a min at x = a. c)Ifh(x) = % for polynomial functions f and g, then h will have a vertical asymptote. d) If you have a max or a min at x = a, then the first derivative = 0 at x = a. e) For a curve to have a horizontal asymptote, evaluate lim f(x) X00 Name: Date: Thinking (12 marks). Show proper steps. Factor Derivatives Fully. 2 x\"2x 1P a) Determine the first derivative and the critical points. 6. Consider the function f(x)= (Computer aids not allowed). b) Determine the second derivative and the inflection points. (More space next page). Name: Date: 6-continued f(x) = * -2x (x + 1) 2 6c) Graph the function. You must label intercepts, max/mins, asymptotes, and Ips. 76 5 + +Name: Date: Application (12 marks) Full solutions are required as indicated at the top of the test. 7. Person A is 4 km south of person B. Person A walks north at a speed of 2 km/h. Person B walks east at a speed of 3 km/h. If they both start at 12:30:00, determine the time on the clock in hours minutes and seconds when they will be closest together. Create/complete the diagram that supports your opening equation. (6 marks) 10 km 4 km Name: Date: 8. A lidless cylinder is to be built to have a volume of 4000 cm?. The material cost to construct the base is triple the cost to build the lateral surface. Determine the dimensions of the cylinder that will produce the lowest cost. Include a well labeled diagram. (6 marks) Answer Step 1 of 2 Solution: Given function curve is derivative of f(x) The given graph is a derivative graph Concave up when the curve is increasing means fr(z) >0 or slope of the curve is positive concave up intervals (o0, 0.5)U(2.5, o) Expalantion Concave down when slope of derivative is negative ,concave up when slope of derivative is positive Step 2 of 2 Concave down function curve slope is decreasing fr(z)