MATH 150A, Review for Midterm Test 1

Test topics

1. Functions and their graphs
   1. Rational functions and asymptotes (later with connection to infinite limits and limits at infinity). Piecewise defined functions (1.1)
   2. Trigonometric functions (Appendix)
   3. New functions from old functions: function transformations and graphs. Composition of functions (1.3)

2. Limits and continuity
   1. The limit of a function: basic examples. Piecewise defined functions. Left and right limits (2.2)
   2. Rational functions; infinite limits and vertical asymptotes; limits at infinity and horizontal asymptotes (2.2)
   3. Computing the limits using limit laws (2.3)
   4. The ε-δ definition of the limit (2.4)
   5. Continuity (2.5)

3. Derivatives
   1. Tangent and velocity problems (2.1)
   2. Derivatives and rates of change (3.1)
   3. The derivative as a function (3.2)
   4. Basic rules for computing derivatives: power rule, constant multiple rule, sum/difference, product/quotient (3.3)
   5. Derivatives of trigonometric functions (3.4)
   6. The chain rule (3.5)
   7. Implicit differentiation (3.6)
   8. Applied problems (3.7)
   9. Related rates (3.8)
   10. Linear approximation and differentials (3.9)

Important definitions/facts

• The limit of a function: intuitive definition, and the ε-δ definition
• Continuous function; intermediate value theorem
• Average rate of change (difference quotient) and instantaneous rate of change (the derivative)
• Definition of the derivative using limits
• Differentiable function; examples of non-differentiable functions
• The two remarkable limits for trigonometric functions: (sin x)/x and (1-cos x)/x

List of possible types of problems

• Find the limit of a function: check if a function is continuous, if not, find out if it has an asymptote at the limit point; if not, use algebra to bring to the form in which computational rules for limits may be applied.
• Find the limit using the ε-δ definition: for every ε>0 given, find a δ such that 0<|x-a|<δ implies that |f(x)-L|<ε.
• Find a tangent line to the graph; find a line perpendicular to the graph; find a tangent line to a curve using implicit differentiation.
• Determine where a given function is continuous/discontinuous; find the values of parameters in a piecewise defined function so that it becomes continuous.
• Find the derivative using first principles (the definition as a limit)
• Find the derivative using the algebraic rules (constant multiple, sum, difference, product, quotient, chain...). Simplify enough to be able to compute the value of the derivative at a given point.
• Related rates and other applications: word problems from Sections 3.7 and 3.8.
• Find the linear approximation for a function near a point; use the linear approximation to compute approximate value of a function. Find the differential of the function.

Review problems

General rules

• All tests are closed books/notes; graphing calculators, cell phones, and laptops are not permitted. A basic scientific calculator may be helpful (example: TI-30X IIS or similar) but it is not required to answer the questions
• The length of the exam is 1 hr 5 minutes
• The format of the exam will be similar to that of the quizzes, however it will be proportionally longer, and some of the problems may be more challenging than the quiz problems. You should expect the general level of questions to match that of the homework problems.