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.
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]?
[math]f'\left(x\right)=cos\left(x\right)[/math]
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]?
[math]f'\left(x\right)=-sin\left(x\right)[/math]
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].
[b][color=#9900ff]Hints: [/color][/b] [br][br]1) Recall the identity for the sine of the sum of two angles. (Look it up if you need to!)[br] You can also refer [url=https://www.geogebra.org/m/gNVjYaPy]here[/url]. [br][br]2) It may help you to refer to the first applet displayed within [url=https://www.geogebra.org/m/h4wT8pK3]this worksheet[/url].
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].
[color=#9900ff][b]Hints:[/b][/color][br][br]1) Recall the identity for the cosine of the sum of two angles. (Look it up if you need to!) [br] You can also refer [url=https://www.geogebra.org/m/gNVjYaPy]here[/url]. [br][br]2) It may help you to refer to the 2nd applet displayed within [url=https://www.geogebra.org/m/h4wT8pK3]this worksheet[/url].