Derivatives and Curviness
How to use Derivatives when Describing “
Curvy-ness” of Functions
Most students when they first learn about derivatives in Calculus are not exposed to the many uses of this important topic. However, students have heard the term “derivative” disguised as words like SLOPE and RATE OF CHANGE! So how do we use derivatives when describing how curvy a function is? Well, since derivative is just a fancy word for slope, how about finding the slope at select points to see if we change from a positive value to a negative value, or vice-versa, or if we even change signs at all!
For example, I was tutoring a student who had to find the maximum and minimum of various functions on some closed interval. Well we know that if we have a closed window we are looking into, then at some point the curve we are analyzing will have a highest peak and lowest trough somewhere, even if those points are the endpoints of the specified closed interval. Think about a linear function: the derivative of any linear function will be constant. What does that mean when describing the “curvy-ness” of linear functions? It means the slopes at each point in some closed interval is fixed, i.e. if the derivative is negative, then the slope is negative, then the maximum will be the leftmost end-point and the minimum will the rightmost endpoint. *Note the max is the rightmost endpoint and the min is the leftmost endpoint when the derivative is positive* Now what if we have an exponential or logarithmic function. Well, the same argument applies: if the derivative of either function is positive, then the rightmost point will be higher than the leftmost point; if the derivative of either function is negative, then the opposite is true. Why is this true? Think if someone pulled a linear function with some slope (other than 0) straight in either the x direction or the y direction; now we have a bend in our curve, yet we will still have a positive or negative slope AT EVERY POINT!
Now let’s talk about some more interesting functions. What about wiggly and curvy functions??? A wiggly function, like sin(x) or cos(x), will oscillate between some y-axis interval. Meaning we have a maximum y-value and a minimum y-value between some closed x-axis interval. The question is WHERE? We can no longer assume the max/min will be at the endpoints. Remember a term called “critical values”? I can almost hear the moans and groans bringing this term up again, but do not fret. Basically, we need to know when the signs of our slope changes. The method? You guessed it! Take the derivative, and this time set that derivative equal to zero. Why? We want to find out when our function has zero slope. Think about that moment at the very peak of your favorite roller coaster: there is a very brief moment when the machine no longer has to trek your car up a very steep slope, and you are motionless as you look upon the horizon straight in front of you. Or when your car starts its trudge down the other side of this steep slope, down the pits of gravity when all weight is thrusted perpendicular on top of your shoulders into the seat of your car. This is the very most bottom of the hill, at least for that section of the roller coaster, i.e. some interval of this curvy coaster! So the same argument applies for polynomials of some high degree, where the function wiggles many times in some interval. Just find the places where we have zero slope and calculate whether we have positive or negative slopes BETWEEN our critical values and voila! We know where our peaks and troughs are, and therefore we know the “curvy-ness” of any function.
All this can be analyzed WITHOUT graphing these functions at all! Pretty cool right?