Two posts in one week messes up my naming system.
This post is about derivatives and epiphanies about derivatives. Last Monday, 10th of October, we did an activity in class that had us explore transformations of functions and their effects on the derivatives of those functions. It turns out that reflections and vertical stretches affect both the function and its derivative. For example, say f(x)=x^(1/2) with a derivative of f'(x)=1/2*x^(-1/2). If f(x) is vertically stretched by a factor of 2 and it is reflected over the x-axis, the new function would be f(x)=-2x^(1/2). What we learned in this activity is that the transformations done to the function will be the same transformations done to the derivative. The derivative of the new function is f'(x)=-x^(-1/2), which is the derivative of the original function vertically stretched by a factor of 2 and reflected over the x-axis. Although in the activity we only did this with square root functions, this property applies to all types of functions because square root functions can be treated like any other exponential function. For example, if g(x)=2x^3 and it is vertically stretched by a factor of 3 to become g(x)=6x^3, then its original derivative, g'(x)=6x^2 will become g'(x)=18x^2. The new derivative is the old derivative vertically stretched by a factor of 3. I tried to make a GIF of a graph demonstrating this concept, but that failed miserably; I have instead linked it.
This post is about derivatives and epiphanies about derivatives. Last Monday, 10th of October, we did an activity in class that had us explore transformations of functions and their effects on the derivatives of those functions. It turns out that reflections and vertical stretches affect both the function and its derivative. For example, say f(x)=x^(1/2) with a derivative of f'(x)=1/2*x^(-1/2). If f(x) is vertically stretched by a factor of 2 and it is reflected over the x-axis, the new function would be f(x)=-2x^(1/2). What we learned in this activity is that the transformations done to the function will be the same transformations done to the derivative. The derivative of the new function is f'(x)=-x^(-1/2), which is the derivative of the original function vertically stretched by a factor of 2 and reflected over the x-axis. Although in the activity we only did this with square root functions, this property applies to all types of functions because square root functions can be treated like any other exponential function. For example, if g(x)=2x^3 and it is vertically stretched by a factor of 3 to become g(x)=6x^3, then its original derivative, g'(x)=6x^2 will become g'(x)=18x^2. The new derivative is the old derivative vertically stretched by a factor of 3. I tried to make a GIF of a graph demonstrating this concept, but that failed miserably; I have instead linked it.