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′−aThe quadratic approximation of a function at (near) x=a is: (x)Q(x) =f(a) +f′(a) (x)−a+2f(a)′′−a2A 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 ato 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−af(b)−f(a)a<c<bIn 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 Pointsare when f’(x)=0 or DNE (and are in the domain of f(x)) Local Extremalocal( 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.