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Midterm Exam Prep Seminar Oct. 25
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Try a Free Sample of Grant's Audio Lectures
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Don't have my book or audio? You can download a free sample of my book and audio lectures containing Lessons 1 and 2:
Did you read my tips on how to study and learn Math 1500? If not, here is a link to those important suggestions:
These are tips for the first assignment in the Distance/Online Math 1500 course, but I strongly recommend that you do this assignment as homework even if you are taking the classroom lecture section of the course. These assignments are very good (and challenging) practice.
The first assignment is a great way to build and review key skills that will be helpful for this course. Here is a link to the actual assignment, in case you don't have it: You need to study Lesson 2 (Limits), Lesson 3 (Continuity), Lesson 4 (The Definition of Derivative), and Lesson 5 (The Differentiation Rules) from my Intro
Calculus book to prepare for this assignment.
Classic continuity question like my Lesson 3, questions 1 to 3. Make sure you use the correct piece for f(2), limit as x approaches 2-, and limit as x approaches 2+. Hint, each piece will be used once, and only once.
Don't let
the trig and logarithm scare you. Remember that I do a quick trig review in Lesson 1 of my book and also in Lesson 2 prior to question 16. I also do a Log and Exponential review in Lesson 1. All of these things are just a matter of subbing in the number. They solve themselves.
This uses the Intermediate Value Theorem like my Lesson 3, questions 4 and 5. First, pull everything over to the left side of the equation, and define the left hand side as your function f(x). Note that f(x) is not a polynomial because of the trig
function, but since sine is a continuous function, you can declare f(x) is continuous. Then prove f(x) has at least one zero on (0,1). Be sure that you say "by Intermediate Value Theorem" as your justification, and make sure you state that f(x) is continuous.
These are rather challenging infinity limits like I teach in Lesson 2.
Part (a) Note the e^x part of this limit is quite simple. See my section about Graphing Exponential Functions in
Lesson 9. The rest is a standard infinity limit like my Lesson 2, questions 10 and 11.
Part (b) Similar to my Lesson 2, question 12. Note that the square root of x^6 is NOT x^3, it is the absolute value of x^3 or |x^3|. Just like I discuss in question 12, that could be x^3 or -x^3, depending. Make sure you write a note, perhaps right at the start of the
question, saying what square root of x^6 simplifies to in this problem to justify your work.
Part (c) See my Lesson 2, Practise Problem 74 for a similar example. Again, include a note about square root of x^2 this time.
Look at my Lesson 2, question 15 to understand the concepts here. - You must find all the bottom zeroes of this function, then solve the limits as x approaches those zeros to find the vertical asymptotes. Make
sure you say, "Vertical Asymptote at x=..." in your conclusion.
- You must compute the limits as x approaches infinity and negative infinity to find the horizontal asymptotes. You must do both limits and they do not have to agree. Each limit looks for its own asymptote. There could be one, two, or no horizontal asymptotes. Make sure you say, "Horizontal Asymptote at y=..." in your
conclusion.
Classic defintion of derivative question. Similar to my Lesson 4, question 2(a). Make sure you read the section on Simplifying Triple Deckers.
Classic differentiation rules practice as I teach in Lesson 5.
Read my section on Velocity and Acceleration starting on page 149. Look at my Lesson 5, question 5 and Practise Problems 86 and 87.
Part (b)
The particle stops rising when its velocity
is 0. Find t where v=0.
Part (c) The particle hits the ground when h=0. Find t when h=0. If you do it correctly, there will be two answers for t, but one of those answers will clearly be inappropriate. That is because the particle is on the ground twice, once at the start, and once at the end.
Similar to my Lesson 5, question 6 and Practice Problem 88. You can use quotient rule, keeping in mind that the derivative of f(x) is simply f'(x) and the derivative of g(x) is simply g'(x). Don't miss the coefficient
rule!
Classic tangent line application of derivatives. Like my Lesson 5, question 2 and Practice Problems 75 to 82.
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