MATH 150A, Review for Midterm Test 2

Test topics

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)

Important definitions/facts

• Applications of calculus to graphing: critical points, local maxima/minima, inflection points, concavity upward/downward
• Formula for Newton's method
• Antiderivative. The indefinite integral
• Area function; definition of the definite integral using Riemann sums
• Properties of integrals: additivity, algebraic operations, inequalities
• Integrals of even and odd functions
• The Fundamental Theorem of Calculus, parts I and II
• The Net Change Theorem and applications to volumes, concentrations, masses and distances (p. 327)
• The substitution rule
• Formulas for the area

List of possible types of problems

• 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)
• Problem on rectilinear motion (Section 4.9). Find distance traveled using integration (Integral with absolute value of the function -- Section 5.4)
• Approximate the integral using a Riemann sum (left end point, right end point or midpoint rules (Section 5.1)
• Find the value of an integral by evaluating the limit of Riemann sums (Section 5.2)
• Compute a definite integral, using elementary geometry (areas of rectangles, triangles, circles (Section 5.2)
• Compute the derivative of a function given as an integral with variable upper or lower limit (Section 5.3)
• Compute antiderivative (indefinite integral), using basic rules or substitution
• Compute a definite integral, using basic rules or substitution
• Sketch a region between two curves and find its area (Section 6.1)

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.