Math 1151, Spring 2024
Written Homework 6
Assignment Goals: The Fundamental Theorem of Calculus tells us about the relationship between the definite integral and the derivative. This assignment will help you to practice:
• Using the Fundamental Theorems of Calculus,
• Using velocity and speed to find displacement and distance traveled, and
• Evaluating complex integrals using substitution.
Problem 1: Using the Fundamental Theorems (24 points)
Directions: Evaluate each of the following. Explain and show your work. You do not need to simplify your answers.
Instructor Note: These problems are all very similar looking, but are actually all very diferent. Understanding these small details is key to using the fundamental theorems of calculus.
(Hint: No part of this problem requires you to calculate the derivative of )
a) (6 points)
b) (6 points)
c) (6 points)
d) (6 points)
Problem 2: Velocity and Speed (20 points)
Directions: Suppose that an object moving along a straight line has position function s(t) = (t2 - 9t + 19)et-1 in feet, with t > 0 measured in minutes. Use this position function to solve the following problems.
Instructor Note: Position, velocity, and speed were some of the motivating inluences in the early days of calculus.
a) (2 points) Show that the velocity function, v(t), is equal to v(t) = (t2 - 7t + 10)et-1.
Since the answer for this part is given, it is especially important to show each step in your work.
b) (4 points) Write a deinite integral that gives the displacement of the object during the time interval [1, 4]. Then calculate the displacement.
c) (3 points) Write a deinite integral which gives the average velocity of the object over the time interval [1, 4]. Then ind its value.
d) (8 points) Write a deinite integral that gives the total distance traveled of the object during the time interval [1, 4]. Then calculate the distance traveled.
e) (3 points) Write a deinite integral which gives the average speed of the object over the time interval [1, 4]. Then ind its value.
Problem 3: Integration via the Substitution Method (16 points)
Directions: Evaluate each of the following. Explain and show your work. You do not need to simplify your inal answer.
Instructor Note: Substitution is one of the fundamental methods for evaluating complicated integrals. This is your chance to practice writing out and explaining the method. You can use the feedback you get to correct any issues before the inal exam. Don’t be afraid to experiment with your choice of substitution - the best way to learn is by trying things out!
In this homework we give hints about a proper choice for substitution. On exams you will usually not be given this hint - it will be up to you to make a useful choice for the substitution. It’s a good idea to include this decision making as part of your study routine for this topic.
a) (8 points) 2 sin(t) cos(t) [cos(t) + 1]2 dt [ Hint: Try using the substitution u = cos(t).]
b) (8 points) [ Hint: Try using the substitution v = u - 2.]