Calculus

1) A gas station sits at the intersection of a north-south road and an east-west road. A police car is traveling toward the gas station from the east, chasing a stolen truck which is traveling north away from the gas station. The speed of the police car is 100 mph at the moment it is 3 miles from the gas station. At the same time, the truck is 4 miles from the gas station going 80 mph. What is the rate of change of the straight line distance between the two vehicles at this moment?

4. How accurate is this approximation? Use the Remainder Estimation Theorem to determine the upper bound for 102/3 - P₂(10)| = |R₂(10)|.

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.

1. ) Consider a set of numbers {ai} define as ai-3-3i for integers i such that i > 1. Use the properties of sums in THEOREM 5.3.7 to evaluate the following sums.

4. (§5.2, 5pts each) Each function below represents the velocity (in ft/s) of an object which is thrown straight up at some initial height. Use each function to determine: • What is the object's initial velocity? When does the object land on the ground? How long does it take for the object to return to its initial height? What is the maximum height of the object? Remember to include the proper units for your final answers. Round any answers to two decimal places if needed. a. v(t) = -32t+42 with initial height 10 ft b. v(t) = -32t +78 with initial height 70 ft

1. Find an equation of the tangent plane to the surface z = y cos(x - y) at the point (3,3,3).

2. Use Lagrange multipliers to find the point on the paraboloid z = x² + y² which is closest to the point (3, -6, 4). Then calculate the perpendicular distance (minimum distance) from the point (3, -6, 4) to the paraboloid.