class: center, middle, inverse, title-slide .title[ # Qualitative Description of a Function via its derivative ] .subtitle[ ## Session 02 ] .author[ ### Alejandro Ucan ] .date[ ### 2022-10-15 ] --- # Goals: * Describe the function via its derivative. <br/><br/> * Compute: extreme values, monotony intervals, and concavity intervals. <br/><br/> --- # How the Derivative is related to the function? <iframe src="https://www.desmos.com/calculator/2ew0jangf3?embed" width="1200" height="450" style="border: 1px solid #ccc" frameborder=0></iframe> --- # Extreme Values: > To find the extreme values of a function, we need to search for those values of `\(x\)` where the derivative is cero. These points are called __critical__ points and satisfy `$$f'(x)=0.$$` <br/><br/> > __First derivative criterium:__ A critical point `\(x_0\)` is said to be: * a _minimum_ of the function if `\(f'(x)<0\)` for `\(x<x_0\)` and `\(f'(x)>0\)` if `\(x>x_0.\)` <br/><br/> * a _maximum_ of the function if `\(f'(x)>0\)` for `\(x<x_0\)` and `\(f'(x)<0\)` if `\(x>x_0.\)` --- ## Example: > The function `\(f(x)=-x^2\)` has a maximum in `\(x=0.\)` <iframe src="https://www.desmos.com/calculator/hlq2dsx9yj?embed" width="1200" height="300" style="border: 1px solid #ccc" frameborder=0></iframe> --- ## Example: > `\(f'(x)=-2x\)` <br/><br/>. Which are the critical points? `$$-2x=0\Rightarrow x=0.$$` --- # Monotony. > We say that a function `\(f\)` is: * _increasing_ in an interval `\((a,b)\)` if `\(f'(x)>0\)` for each point of `\((a,b).\)` <br/> * _decreasing_ in an interval `\((a,b)\)` if `\(f'(x)<0\)` for each point of `\((a,b).\)` --- ## Example: Find the intervals where `\(f(x)=x^2-6x+5\)` is monotonus. > Solution: <br/> * Find the critical points (solve for `\(f'(x)=0\)`): `$$f'(x)=2x-6=0\Rightarrow x=3.$$` * We made a test: we check the sign of the derivative at one point in `\(x<3\)` and in other point in `\(x>3.\)` <br/><br/> * Increasing at `\((3,+\infty)\)` and decreasing at `\((-\infty,3).\)` --- # Excercise: Find the intervals where `\(f(x)=2x^3-3x^2-12x+7\)` is decresing. --- # Excercise: Find the intervals where `\(f(x)=2x^3-3x^2-12x+7\)` is decresing. > Solution: <br/> * We compute the derivative `\(f'(x)=6x^2-6x-12.\)` <br/> * Solve for `\(f'(x)<0.\)` `$$6x^2-6x-12<0 \Rightarrow (x-2)(x+1)<0.$$` <br/><br/><br/> * Is decresing at `\((-1,2).\)` --- # Concavity: > We are interested to know where a function looks like a mointain, like a valler and where these types changes. --- # Concavity and derivaties. > For a function `\(f(x)\)`: <br/><br/><br/> * `\(f(x)\)` es __concave up__ at `\((a,b),\)` if `\(f''(x)>0\)` at `\((a,b).\)` <br/><br/> * `\(f(x)\)` es __cóncava down__ at `\((a,b),\)` if `\(f''(x)<0\)` at `\((a,b).\)` <br/><br/> * If `\(f''(x)\)` changes it sign at `\(x=c,\)` then `\(f\)` has an __inflection point__ at `\(x=c.\)` --- # Example: Find the concavity intervals and inflection points of `\(f(x)=2x^3-3x^2-12x+7.\)` > Solution: <br/> * Solve `\(f''(x)=0.\)` $$ f''(x)=12x-6=0\Rightarrow x=\frac{1}{2}.$$ * We made a test for `\(f''\)` for `\(x<1/2\)` y `\(x>1/2.\)` <br/><br/><br/> * It is concave up at `\((1/2,\infty),\)` concave down at `\((-\infty, 1/2)\)` and has an inflection point at `\(x=1/2.\)` --- # Practical excercise: > Determine the monotony intervals, the concavity intervals and inflections points of the following points. Made a sketch of the graph of the function using this information. <br/><br/> 1. `\(f(x)=x^2-12x+1.\)` <br/><br/> 2. `\(f(x)=5-3x^2+x^3.\)` <br/><br/> 3. `\(f(x)=-3x^2+2x-3.\)`