UC Davis MAT 012, Summer Session II, Midterm Examination

Transcription

1 UC Davis MAT 012, Summer Session II, 2018 Midterm Examination Name: Student ID: DATE: August 24, 2018 TIME ALLOWED: 100 minutes INSTRUCTIONS 1. This examination paper contains SEVEN (7) questions and comprises ELEVEN (11) printed pages. 2. This is a closed-book exam. 3. You may use a calculator, but to receive credit you must show all work. 1

2 Exercise 1 Solve the following equations and inequalities t < 3 2. x 2 + x 6 = 0 3. x 3 + 2x 2 3x = 0 2

3 Exercise 2 Consider two functions f(x) and g(x). 1. Describe the domain of the composition f g(x) in terms of the domains of f and g. 2. Suppose f(x) = x + 1 and g(x) = 3x 2. What is the domain of the composition f g? 3. Consider the following transformations: a) What is the function that represents f(x) reflected across the y-axis? Plot this function. b) What is the function that represents g(x) reflected across the x-axis? Plot this function. 3

4 Exercise 3 Suppose (1, 3) and ( 1, 1) are two points on the outside of a circle. 1. What is the distance between these two points? 2. What is the midpoint of these two points? 3. What is the equation for the line that connects the two points? 4. What is the equation for the line that passes through the midpoint, and is perpendicular to the line that connects the points? 4

5 5. Suppose the midpoint of the two points is also the center of the circle. What is the equation for all the points on the circle? 6. What is the equation for all the points on a circle that has the same radius as the circle above, but whose center is the origin? 5

6 Exercise 4 Suppose you are leaning a 3-meter ladder against a wall. (a) Draw a picture that represents the situation. (b) Find the height of the top of the ladder as a function of the distance along the floor between the wall and the base of the ladder. (c) Find the area of the triangle enclosed by the ladder, the wall and the floor, as a function of the distance along the floor between the wall and the base of the ladder. (e) What is the average rate of change of the area of the triangle enclosed by the ladder between when the floor distance is 0 meters and when it is 1 meter? 6

7 Consider the triangle below. Exercise 5 h Suppose that the height h and the base b add to 4. b (a) Write an equation that describes the area enclosed by the triangle as a function of the base b. (Recall that the area of a triangle is base height 2.) (b) What length b maximizes the area of the triangle? (c) What is the maximum area of the triangle? 7

8 Exercise 6 Suppose f(x) = (x 1)(x + 1)(x 2), g(x) = (x 1) 2 (x + 1), and h(x) = (x 1) Graph f(x). Label all x-intercepts. 2. Graph g(x). Label all x-intercepts. 3. Graph h(x). Label all x-intercepts. 4. One of the three functions f, g, and h has an inverse function. Which function is it, and why? Explain your answer either graphically or algebraically. 8

9 5. For the function you identified in the previous part, find its inverse function. 9

10 Suppose f(x) = 1 x and g(x) = 1 x 2. Exercise 7 1. Compare and contrast f and g using the following table. Identify at least 3 similarities and at least 3 differences. (Options include but are not limited to: domain, range, symmetry, asymptotes, turning points, maxima/minima, existence of an inverse,...) Similarities Differences 2. Graph f(x) translated to the left by 2 units. What equation describes this function? 10

11 3. Graph g(x) translated up by 3 units. What equation describes this function? 4. Plot the function y = 4x 3 x+1. Label all asymptotes. 5. Plot the function y = (x 1) 4. Label all asymptotes. 11