Category Archives: Test Review

Test 2 Topics

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You should be able to:

  • Determine whether the limit exists by looking at the graph of . Sec 1.4 
  • Draw the graph of a function in which the limit  does not exist. Sec 1.4 (also, see Test 2 practice problems)
  • Determine whether a function is differentiable at a point by looking at the graph for . Sec 1.5
    • Recall that is differentiable at a point when the following limit exists: exists.
  • Determine whether a function is continuous at a point.
    • Recall that is continuous at a point when the limit exists is equal to the value . Sec 1.5
  • Interpret the derivative of at a point as the (instantaneous) rate of change of . Sec 1.8
    • In populations applications: we call the derivative the growth rate
    • In physical applications: we call the derivative velocity
  • Compute the average rate of change, and compare it to the derivative at a point. Sec 1.8
  • Interpret the second derivative of at a point as the (instantaneous) rate of change of the rate of change of . Sec 1.8
    • In population application: we can the second derivative the rate of change of the growth rate
    • In physical applications: we call the second derivative acceleration
  • Compute the second derivative Sec 1.6-1.7
    • I will specifically test you on using the general power rule. Sec 1.6
  • Given a function compute its intervals of increasing and decreasing Sec 2.2
  • Given a function compute its relative minima and maxima Sec 2.3
  • Given a function compute its intervals of concavity Sec 2.2
  • Given a function compute its inflection points Sec 2.3
  • Use the above information to sketch the graph of  Sec 2.2-2.4
    • You will not be asked to find the and intercepts for .
  • Given information about , , and , at a point sketch the graph of near the point . Sec 2.2
  • Interpret intervals of increasing and decreasing, relative maxima and minima, and concavity in applications (focus on physical applications) See Application of derivative HW and examples from lecture
  • Given information about and (in a graphical form) answer questions about . Sec 2.2 
    • Specifically try numbers 23, 24, 43, and 44 from Sec 2.2; note that number 44 is the test review problem number 4

Good luck!

Test 1 Suggested Problems

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From the text book supplement: ma_131_supplement. Answers to odd questions are posted at the end of the supplement.

  • Sec 10.1: 17, 19, 23
  • Sec 10.2: 1,3, 5, 19
  • Sec 10.3: 1, 3, 5, 7, 15, 17, 19, 25, 27
  • Sec 10.4: 5, 7, 9, 11

From Section 1.1-1.4 in the textbook (exercise numbers are from the 13th edition). Odd answers are in the back of the book.

  • Sec 1.1: 1, 3, 7, 9, 11
  • Sec 1.2: 5, 7
  • Sec 1.3: 1 (remember the derivative of a straight line is just its slope), 3, 5, 7, 9, 11, 13, 15, 17, 19, 23; Find the equation of the tangent line for the function from problems 33, 35, 37, 39, 58 (solution: , and ), 63
  • Sec 1.4: 29, 30 (solution 12), 31, 32 (solution 2)

Test 1 Topics

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Difference equation material can be found in sections 10.1-10.4 of the supplemental text. You can find the supplemental text in the resources section of WebAssign, but I am also posting it here: ma_131_supplement

You should be able to:

  • Write the difference equation for a word problem (applications in both finance and population growth). Sec. 10.1
  • Write the general solution. (Don't forget about the special situation when .) Sec. 10.2
  • Sketch the graph of various difference equations. Sec. 10.3
  • Describe the long term behavior and vertical direction of the graph of a difference equation. Sec. 10.3
    • Sketch the graph of a function that is:
      • monotonic, oscillating, increasing, decreasing, constant, unbounded, bounded (asymptotic to a line).
    • Sketch the graph of a function that is neither monotonic, oscillating, nor constant.
  • Solve a word problem by using the graph of the difference equation (see for example example 10 on page 400 of the supplemental text). Sec. 10.3
    • Practice problem 3 may also be helpful (its solution is given after the homework problems). For the this type of problem, I would like you to include three graphs on your test.
  • Solve a word problem using the general solution. Sec. 10.4
    • You may be asked to solve for an initial deposit or loan amount; or a monthly withdrawal, payment or deposit amount.

You are also responsible for Sections 1.1-1.4 (pages 86-87 on the limit definition of the derivative) from the text book.

You should be able to

  • Graph a linear equation and give its slope. Sec 1.1
  • You should be able to use the power rule to compute the derivative of function at a point or as a function. Sec 1.2
  • You should be able to give the slope of the tangent line for a function at a point either using the power rule or the graph of and . See for example, Quiz 2. Sec 1.3 
  • You should be able to sketch the graph of a tangent line (as we did for ).
  • You should be able to use the limit definition of the derivative to compute the derivative of a function at a point . See for example Example 7 from Section 1.3 and Examples 5-7 from Section 1.4. Sec 1.3-1.4 
    • I will provide the equation for this limit .