MATH 150A, Review for the Final Exam

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)

4. Applications of differentiation
   1. Maximum and minimum values (4.1)
   2. The mean value theorem (4.2)
   3. How derivatives affect the shape of a graph (4.3)
   4. Limits at infinity. Horizontal asymptotes (4.4)
   5. Summary of curve sketching (4.5)
   6. Optimization problems (4.7)
   7. Newton's method (4.8)
   8. Antiderivatives (4.9)

5. Integrals
   1. Areas and distances (5.1)
   2. The definite integral (5.2)
   3. The fundamental theorem of calculus (5.3)
   4. Indefinite integrals and the net change theorem (5.4)
   5. The substitution rule (5.5)

6. Applications of integration
   1. Areas between curves (6.1)
   2. Volumes by the method of slices (6.2)
   3. Volumes by the method of cylindrical shells (6.3)
   4. Mechanical work (6.4)
   5. Average value of a function (6.5)

Important definitions/facts

• The limit of a function: intuitive definition
• Definition of the derivative as a limit of the difference quotient
• Equation of the tangent line. Slope of the tangent line
• Basic rules for computing derivatives: sum/difference, scalar multiple, product, quotient. The chain rule.
• Critical values (points). Relative maxima/minima
• Global maximum/global minimum
• Intervals of increase/decrease. Inflection points. Intervals of concavity upward/downward
• Limits at infinity. Horizontal asymptotes. Oblique asymptotes.
• Formula for Newton's method
• Antiderivative. The indefinite integral
• Rules for computation of integrals: sum/difference, constant multiples, substitution
• Related rates
• Representing integrals as areas (rectangles, triangles, circles...)
• The Fundamental Theorem of Calculus, parts I and II
• Areas between the curves
• Volumes by slices and cylindrical shells
• Mechanical work expressed as an integral
• Average value of a function

List of possible types of problems

• Find the limit of a function: review rules/techniques for computing limits
• Compute a definite/indefinite integral: review rules/techniques for computing integrals
• 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 (problems in Section 2.5)
• Find the derivative using first principles (the definition as a limit of the difference quotient) (examples in Sections 3.1-3.2)
• 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.
• Find the derivative using fundamental theorem of calculus (examples in Section 5.3)
• Related rates and other applications: word problems from Sections 3.7 and 3.8.
• Sketch the graph of a function, finding intercepts, asymptotes, critical points, local and global maxima/minima, intervals of increase/decrease, inflection points and intervals of concavity upward/downward (Sections 4.1-4.5)
• Word problems on optimization (Section 4.7)
• Use Newton's method to approximate a solution of a nonlinear equation (Section 4.8)
• Approximate the integral using a Riemann sum (left end point, right end point or midpoint rules (Section 5.1)
• Compute a definite integral, using elementary geometry (areas of rectangles, triangles, circles (Section 5.2)
• Sketch a region between two curves and find its area (Section 6.1)
• Find a volume of a solid (Sections 6.2, 6.3)
• Compute mechanical work by expressing it as an integral (Section 6.4)
• Use mean value theorem for integrals (problems in Section 6.5)

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 2 hours
• The format of the exam will be similar to that of the midterms. You should expect approximately 10 problems (some with a, b, c, d...). Partial credit will be given depending on the amount of work shown, and its relevance to the solution of the problem.