Practice Midterm 1 Solutions

Like my homework 4 solutions, what you'll find here aren't straightforward solutions, but can be helpful. And I might occasionally throw in other random pitfalls to avoid.

1a) This type of problem is straightforward. The individual functions are continuous at x=0 so their sum is continuous at x=0, and so by definition, the limit of a function f continuous at x=a as x tends to a, is f(a). Problems similar to this will use pg 99 Limit Laws, pg 101 other Limit Laws, pg 102, pg 119, pg 122, pg 124, pg 125.

1b) Read examples about quotients of polynomials. pg 101 example 2. pg 102 example 3. pg 133 theorem 5. pg 133 example 3. pg 134 example 4. pg 135 example 5.

Important: pg 133 states that most of the limit laws in section 2.3 also hold for limits at infinity. Read this paragraph.

It's important to read other examples in section 2.3 and other examples in section 2.6 as well.

1c) This definitely uses theorem 8 on page 125. So first find the limit of the expression inside arcsin. One can rationalize the numerator, or factor the denominator as done on page 125 example 8.

1d) This is, like 1c, similar to an example from the book. Check out page 135 example 5.

Important: read examples from section 2.3 and examples from 2.6.

2) Know the definition of horizontal and vertical asymptotes (pages 132 and page 95, respectively).

Note: Understanding the definition of a vertical asymptote, it is enough to have either a left-hand limit at x=a or a right-hand limit at x=a go to infinity, to say x=a is a vertical asymptote of a function f(x).

TYPO: I didn't catch this earlier, but this problem has an error.
The domain is incorrectly written

It should somehow indicate f(x) does not exist at x = 1.
or


3a) Explain why this function is not one-to-one and hence is not invertible. Read page 61, the definition of an inverse function carefully.

3b) Go through the process of finding the inverse function, but remember to indicate the domain of the inverse function. Again, read the definition of an inverse function on page 61 carefully. If f has domain A and range B, then its inverse has domain B and range A.

Note, page 61 warns the reader to not confuse notation. Do not mistake
and

If for some reason you get confused on the exam, feel free to clarify with whoever is proctoring.

4) This problem has multiple parts for which you can get credit.
First, you should be able to identify that the slope of the tangent line must be 3/2.
Second, you should take the derivative of f(x) (here use the power rule, but the problem can give you any function you've learned how to derivate using material covered in weeks 1 through 5.
Third, solve for x's for which f'(x)=3/2.
Fourth, the answers to the first question are of the form (x,f(x)). In this problem, there is only x for which f'(x)=3/2 and so there is one point.
Fifth, to write the equation of the normal line through this point, you must realize that you need to take -1/m where m is the slope of the tangent line (3/2). Hence the normal line has slope -2/3.
Sixth, use the point-slope formula to determine the equation of the normal line, using the slope -2/3 and the point (x,f(x)).

Note: Typically, even if you got a value wrong, if you correctly carry the value over to the next part of the problem, you will get credit. If you cannot obtain values, you should still write down the formula you would have used. In this case, the slope of a normal line is -1/m and the point-slope formula are relevant formulae that might help you get partial credit.

On that note, you should have an intuitive idea of what you can and can't argue credit for. If you felt like you sufficiently showed you knew what you were doing and didn't get enough credit for a problem, then by all means ask. More often than not, it doesn't hurt to ask. A popular reason that credit won't be awarded to you is when everybody else who put the same work you put into the problem, got the same amount of credit you did. Made-up example: Nobody who wrote the point-slope formula got a point for writing the point-slope formula. Nevertheless, I recommend showing you know what you're talking about when possible (as well as hiding what you don't know you're talking about when possible.)

Finally, show work and/or explanation even when a problem doesn't explicitly ask for it. While you may be marked off for extra information which is incorrect, you won't be marked off for extraneous relevant and correct information. You don't have to quote theorem number and pages, but when possible, state relevant steps. Examples: "because the function is continuous" "the limit of a sum is the sum of the limits" "by definition of a vertical asymptote".