Calculus

Instructions: Do all of your work on this paper. Scan the completed problems and create one PDF of your work. Your PDF needs to be at least the same number of pages as the original docu- ment. Upload the PDF to this assignment in GRADESCOPE by the due date. (The Gradescope assignment can be accessed through Canvas.) (1) Let f(x) = cos²(x) sin(x²). Determine f'(x).

Test 5 MAT 175 Name: Read each question carefully. Make sure answers are simplified, exact, and in the correct form, especially to word problems, using complete sentences and correct units. Good luck! 1. (21) For the function f(x)=x²-x+2, Chapter 5 a) Evaluate the Riemann sum for f(x) for the interval 1sxs3 with four subintervals, taking the sample points to be right endpoints. b) Use the definition of a definite integral (with right endpoints) to calculate the value of the integral f(x)dx. (This means the use of summations and a limit) c) Use the Evaluation Theorem to check your answer to part b). Show your work.

1. (a) Differentiate y = sin(22) cos(2x). (b) For g(x) = 3x² loge (x² + 1), find g'(x).

2. Consider the function, The function f(x) = 2x³ + ax² − bx +3 has a factor (x + 3). When f(x) is divide by (x - 2), the remainder is 15. (a) Show that a = 3 and b = 8. (b) Find the other two factors of f(x). (c) Find axes intercepts for f(x). (d) Use calculus to find x-coordinates of turning point for f(x). Use √57 = 7.55 I (e) Sketch the graph of y = f(x). (3 marks) (3 marks) (2 marks) (3 marks) (3 marks)

1. (§5.1, 3pts each) Evaluate each of the following indefinite integrals.

3. (§5.2, 5pts each) Each graph below represents the velocity function of an object moving in a straight line. For each graph, determine the following: • What is the object's maximum velocity? What is the object's maximum displacement? What is the object's total displacement? Remember to include the proper units for your final answers.

The object's velocity at time t is The object's speed at time t is The object's acceleration at time t is (Simplify your answer.) An object is dropped from a tower, 186 ft above the ground. The object's height above ground t sec into the fall is s = 186 - 16t². a. What is the object's velocity, speed, and acceleration at time t? b. About how long does it take the object to hit the ground? c. What is the object's velocity at the moment of impact? ft/sec. ft/sec² It takes sec for the object to hit the ground. (Round to the nearest tenth.) The object's velocity at the moment of impact is (Round to the nearest tenth.) Question 1 of 20 > ft/sec. This This

(6) Olivia is vacationing in a cabin along a river. The path of the river near Olivia's cabin can be modeled by the function y = r². Olivia's cabin is located at the point (0,3) on the diagram. He noticed that a boat is floating at the point (3,2.25) on the river. River 2 Cabin (a) What is the distance of the boat from Olivia's cabin? (3.2.25) (b) What is the minimum distance from Olivia's cabin to the river? (c) What are the location(s) on the river that are closest to Olivia's Cabin? Page 3 of 4

Problem (7) (12 points) Given function y = = f(x) =−x² + 2x + 15. (a) Find f'(x) by the limit definition. (b) Sketch the graph of y = f(x) (include the z- & y- intercepts and the vertex). (c) Find the equation of the tangent line at x = 2. (d) Find the area of the region enclosed by the curve y = f(x), x=0, x= 1, and the z-axis. [Hint: the graph you have sketched in part (b) may help.]

2. A ladder 5 m long is leaning against a vertical wall. If a person pulls the bottom of the ladder away from the wall at a rate of 0.5 m/s, how fast is the top sliding down the wall when the bottom of the ladder is 3 m from the wall?