Calculus 1 Lecture Notes by dFbSGbF

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									Calc 1 Lecture Notes                   Section 3.6                            Page 1 of 3

Section 3.6: Overview of Curve Sketching
Big idea: You can tell a lot about the graph of a function using algebra and calculus skills.

Big skill: You should be able to sketch a fairly accurate graph of any function using techniques
from algebra and differential calculus.

Steps to take when sketching a function:
    I. Algebra Skills
           1. Identify the domain of the function.
           2. Identify x and y intercepts; use Newton’s Method, if necessary. This may involve
           factoring a numerator.
           3. Identify any vertical asymptotes. This may involve factoring a denominator.
           4. Identify asymptotes as x  ±∞.
   II. Calculus Skills
           1. Calculate the first derivative; find critical points.
                    i. Extrema will be at points where the derivative is zero.
                   ii. Increasing and decreasing intervals correspond to positive and negative
                       derivatives, respectively.
                  iii. Vertical tangent lines may be at points where the derivative is undefined.
                  iv. Extrema may also be at points where the derivative is undefined, like for
                       the absolute value function.
           2. Calculate the second derivative; find critical points again.
                    i. Inflection points will be at points where the second derivative is zero.
                   ii. Concave up and down intervals correspond to positive and negative
                       second derivatives, respectively.
Calc 1 Lecture Notes                   Section 3.6   Page 2 of 3

Practice:
                     x4  x2  6
Sketch f ( x) 
                  x3  3x 2  3x  3
Calc 1 Lecture Notes            Section 3.6   Page 3 of 3


Sketch f ( x)  2 x  sin 2 x

								
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