Give examples to show that a cubic polynomial can have zero, one, or two critical points. How many and what types of local extremes are there? Collapse menu 1 Analytic Geometry 1. Lines 2. Distance Between Two Points; Circles 3.
How to Find Local Extrema with the First Derivative Test
Functions 4. The slope of a function 2. An example 3. Limits 4. The Derivative Function 5.
The Power Rule 2. Linearity of the Derivative 3. The Product Rule 4. The Quotient Rule 5. The Chain Rule 4 Transcendental Functions 1. Trigonometric Functions 2. A hard limit 4. Derivatives of the Trigonometric Functions 6. Exponential and Logarithmic functions 7.
Derivatives of the exponential and logarithmic functions 8. Implicit Differentiation 9. Inverse Trigonometric Functions Limits revisited Hyperbolic Functions 5 Curve Sketching 1. Maxima and Minima 2. The first derivative test 3. The second derivative test 4. Concavity and inflection points 5. Optimization 2. Related Rates 3.
Newton's Method 4.
Linear Approximations 5. The Mean Value Theorem 7 Integration 1. Two examples 2. The Fundamental Theorem of Calculus 3. Some Properties of Integrals 8 Techniques of Integration 1. Substitution 2. Powers of sine and cosine 3. Trigonometric Substitutions 4.
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Integration by Parts 5. Rational Functions 6. Numerical Integration 7. Additional exercises 9 Applications of Integration 1.
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Area between curves 2. Distance, Velocity, Acceleration 3. Volume 4. Average value of a function 5. Work 6. Center of Mass 7. Kinetic energy; improper integrals 8. Probability 9. Arc Length Polar Coordinates 2. Slopes in polar coordinates 3. Areas in polar coordinates 4.
The turning points of a graph
Parametric Equations 5. Calculus with Parametric Equations 11 Sequences and Series 1. Sequences 2. Series 3. The Integral Test 4. Alternating Series 5. Comparison Tests 6. Absolute Convergence 7. The Ratio and Root Tests 8. Take your number line, mark each region with the appropriate positive or negative sign, and indicate where the function is increasing and decreasing. Theresult is a so-called sign graph for the function. The function switches from increasing to decreasing at —2; in other words, you go up to —2 and then down.
Maximum and minimum values - An approach to calculus
So, at —2, you have a hill or a local maximum. Conversely, because the function switches from decreasing to increasing at 2, you have a valley there or a local minimum. Obtain the function values in other words, the heights of these two local extrema by plugging the x- values into the original function. Thus, the local max is located at —2, 64 , and the local min is at 2, — To use the First Derivative Test to test for a local extremum at a particular critical number, the function must be continuous at that x- value. Check out the brand new podcast series that makes learning easy with host Eric Martsolf.
Trig Identities for Pre-Calculus Of course you use trigonometry, commonly called trig, in pre-calculus. And you use trig id Pre-calculus involves graphing, dealing with Load more.