Vtl is when dydx is not defined when the denominator

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VTL is when dy/dx is not defined (when the denominator =0) * Plug your solutions back into the original curve to find the point where HTL or VTL occurs (or if it does not) Finding tangent line at a point. Find the derivative Plug that point into the derivative Plug point into the original function and solve Write out point slope form 3.9 Linear Approximation (and Quadratic Approximation) The linear approximation of a function at (near) x=a is: ( a )( x ) L ( x ) = f ( a ) + f a The quadratic approximation of a function at (near) x=a is: ( x ) Q ( x ) = f ( a ) + f ( a ) ( x ) a + 2 f ( a ) ′′ a 2 A linear approximation for f(x) near x=a is an overestimate if f’’( a ) < 0 (f is concave down)
is an underestimate if f’’( a ) > 0 (f is concave up) if f’’( a ) = 0, look around a to see if it is concave up or down Linear approximation for a function is the tangent line to that function at x = a. Linear approx. can be used to estimate values of f(x) near x = a. 3.10 Mean Value Theorem Mean Value Theorem: Hypothesis: If a function is continuous on [a, b] AND differentiable on (a, b), Conclusion: Then for f ( c ) = b a f ( b )− f ( a ) a < c < b In other words, there is at least one point between a and b where the derivative is equal to the slope of the line connecting the points a and b. True/false If hypothesis is true, the conclusion is always true If hypothesis is false, the conclusion could be true or false Chapter 4 4.1 Critical Points, Local Extrema, and Inflection Points Critical Points are when f’(x)=0 or DNE (and are in the domain of f(x)) Local Extrema local( minimum or local maximum) given a graph given a function (or its derivative/2nd derivative) Using the 1st Derivative test Find first derivative Set equal to zero and solve for x or solve for when it is DNE Plot the x values found in step two on a number line Make sure to write “f’(x)” on the number line for justification Choose numbers greater than and less than each x value and plot them on the line Plug the chosen numbers into the first derivative and determine if it is positive or negative.
Make sure to show your work when determining positive or negative and put the pluses and minuses on the number line where they belong. If the x value falls between a negative on the left side and a positive on the right side, it will be a local min If the x value falls between a positive on the left side and a negative on the right side, it will be a local max Using the 2nd Derivative test Find first derivative Set equal to zero and solve for x or solve for when it is DNE Find second derivative Plug the x values found in step two, into the second derivative and determine if the outcome is positive or negative Make sure to show your work with either the exact value or with pluses and minuses If the second derivative comes out to be negative, the critical point is a local max If the second derivative comes out to be positive, the critical point is a local min given information in a table or about the function, its derivative, or its second derivative.