Given another calculus exploration paper, we explored how changes to a function relate to its tangent line. We were given the function f(x)=sqrt.x. At the point x=1, the tangent line is y=1/2x+½. We changed the original function, and looked at how this change affected the derivative at the point x=1. The change to the original function was -f(x). The equation of the tangent line at x=1 of this function is -1/2x-½. The change to the function was -f(x). By multiplying the original tangent line by -1, we got the new tangent line as well. The purpose of the first part of this sheet was to allow us to realize that a point’s tangent line will have the same transformation as the function itself.
The second section of the worksheet focused on the derivative of an entire function, as opposed to a single point. Much like the first section, we transformed an original function and studied how how this transformation effected the derivative function. It turns out that the same rules apply to entire derivative functions as they do to single points. We only tested this for square root functions.
I have tested this rule on the function x^3 -2x^2+1=f(x). The derivative of f is 3x^2-4x. If we transform the graph of f by a vertical stretch of 3, (3f(x)), it's new derivative becomes 9x^2-12x. Now if we multiply f' by 3, we get the same function. This is useful and saves us an entire step on fishing derivatives of functions.
The second section of the worksheet focused on the derivative of an entire function, as opposed to a single point. Much like the first section, we transformed an original function and studied how how this transformation effected the derivative function. It turns out that the same rules apply to entire derivative functions as they do to single points. We only tested this for square root functions.
I have tested this rule on the function x^3 -2x^2+1=f(x). The derivative of f is 3x^2-4x. If we transform the graph of f by a vertical stretch of 3, (3f(x)), it's new derivative becomes 9x^2-12x. Now if we multiply f' by 3, we get the same function. This is useful and saves us an entire step on fishing derivatives of functions.