Download Math 171 Final Exam Review: Things to Know

Math 171 Final Exam Review: Things to Know
1. Limits (Chapter 2)
• Know the definition of a limit.
• Recall that lim f (x) = L if and only if lim+ f (x) = lim− f (x) = L.
x→a
x→a
x→a
• Know how to find limits if the graph of a function is given.
• Limit laws
• Compute limits algebraically.
• Know the overall behavior of the following functions: sin x, cos x, tan x, ex , ln x,
1
.
x
• Infinite Limits (These are associated with vertical asymptotes.)
• Limits at Infinity (These are associated with horizontal asymptotes.)
• Continuity
− Know how to determine whether or not a function is continuous at a point.
(See the continuity checklist on page 99.)
• L’Hôpital’s Rule (Section 4.7)
− Know the different indeterminate forms.
− Remember that l’Hôpital’s rule is only applicable to limits that have the fol0 ∞
∞
lowing indeterminate forms: ,
,− .
0 ∞
∞
2. Derivatives (Chapter 3)
• Find the equation of the tangent line of a function at a specified point.
• Recall that differentiability implies continuity (but not vice versa).
• Know what it means for a function to fail to be differentiable at a point. (See page
140.)
• Differentiation Rules
• Derivatives of Trigonometric Functions
− You should know the derivatives of sin x, cos x, tan x, and sec x.
− Know the trigonometric limits on page 163.
• Know the relationship between displacement, velocity, and acceleration. Recall
that speed is the absolute value of velocity.
• Average Velocity and Instantaneous Velocity
• The Chain Rule
• Implicit Differentiation
• Logarithmic Differentiation
• Derivatives of Inverse Trigonometric Functions
− You should know the derivatives of sin−1 x and tan−1 x.
• Related Rates
3. Integrals (Chapter 5)
• Antiderivatives (Section 4.9)
• Left and Right Riemann Sums
• Definite and Indefinite Integrals
• The Fundamental Theorem of Calculus
− Net Area
• Even and Odd Functions
• The Average Value of a Function
• The Substitution Rule
4. Applications of the Derivative (Chapter 4)
• Maxima and Minima
− Local Extrema and Absolute Extrema
− Find critical points.
− Extreme Value Theorem (page 238)
− Know how to locate absolute extrema on a closed interval. (See page 241)
• Test for Intervals of Increase and Decrease (page 246)
• First Derivative Test (page 249)
• Concavity and Inflection Points
− See the test for concavity on page 252.
− IMPORTANT: Inflection points need not occur at critical points. Moreover,
f ′′ (c) = 0 does not imply that f has an inflection point at x = c. You must
justify this by applying the concavity test.
• Second Derivative Test for Local Extrema (page 254)
− Advice: Since the second derivative test is much more efficient than the first
derivative test, you should always attempt the second derivative test for determining local extrema at a critical point. If the second derivative at the critical
point is zero, the test is inconclusive so you must use the first derivative test.
• Curve Sketching
− Given information about f , f ′ , and f ′′ over specified intervals, you should be
able to sketch the graph of f that satisfies the given properties. (Study Graded
Lab #5.)
• Optimization Problems
− Study the guidelines for optimization problems on page 272.
• Linear Approximation and Differentials
• The Mean Value Theorem
− Know the conditions that must be met in order for the theorem to hold.