Derivative of Sine & Cosine Functions (Quick Investigation)
In the applets below, graphs of the functions [math]f\left(x\right)=sin\left(x\right)[/math] and [math]f\left(x\right)=cos\left(x\right)[/math] are shown. [br]In each applet, drag the BIG WHITE POINT along the graph of the displayed function. [br][br]The y-coordinate of the point being traced out = the slope of the tangent line to the graph of f. [br]Interact with each applet for a few minutes, then answer the questions that follow.
1)
Based on your observations, if [math]f\left(x\right)=sin\left(x\right)[/math], can you write an expression for [math]f'\left(x\right)[/math]?
2)
Based on your observations, if [math]f\left(x\right)=cos\left(x\right)[/math], can you write an expression for [math]f'\left(x\right)[/math]?
3)
Use the limit-definition of a derivative to prove that if [math]f\left(x\right)=sin\left(x\right)[/math], then [math]f'\left(x\right)=cos\left(x\right)[/math].
4)
Use the limit-definition of a derivative to prove that if [math]f\left(x\right)=cos\left(x\right)[/math], then [math]f'\left(x\right)=-sin\left(x\right)[/math].