The derivative of [math]f[/math] at [math]x=a[/math] is the slope of the tangent line to [math]f[/math] at [math]x=a[/math]x = a, and is denoted [math]f'\left(a\right)[/math][br]The graph below is interactive. You can change the function, and the x-value for a, and the value of h. [br]To understand the lesson correctly, follow these instructions. [br][list=1][*]Click on "First Point". This is the point at which you want to find the derivative.[/*][*]Click on "Second Point". This is a variable point that will help you find the derivative[/*][*]Click on "Secant Line". This is the line through [math]\left(a,f\left(a\right)\right)[/math] and [math]\left(a+h,f\left(a+h\right)\right)[/math]. Write the slope formula for the secant line on a piece of paper (because you will need it later)[/*][*]Move "h" so that it "approaches 0". Write the limit expression involving what you wrote in part 3 and "h approaching 0"[/*][*]Move "a" around and take note of the behavior of the slope of the tangent line. Where is it positive? Where is it negative? Where is it 0?[/*][*]Now change the function to [math]f\left(x\right)=x^3[/math] and repeat #5. [/*][*]Change the function to [math]\sin\left(x\right),e^x,\ln\left(x\right),\left|x\right|[/math] and repear #5.[/*][/list]
Using the graph of [math]f\left(x\right)=x^2[/math], which of the following statements are true about [math]f'\left(3\right)[/math]
Using the graph of [math]f\left(x\right)=x^2[/math], which of the following statements are true about [math]f'\left(-3\right)[/math]
Using the graph of [math]f\left(x\right)=x^2[/math], which of the following statements are true about [math]f'\left(0\right)[/math]
Using the graph of [math]f\left(x\right)=x^3[/math], which of the following statements are true about [math]f'\left(3\right)[/math]
Using the graph of [math]f\left(x\right)=x^3[/math], which of the following statements are true about [math]f'\left(-3\right)[/math]
Using the graph of [math]f\left(x\right)=x^3[/math], which of the following statements are true about [math]f'\left(0\right)[/math]
Using the graph of [math]f\left(x\right)=\sin\left(x\right)[/math], which of the following statements are true about [math]f'\left(\frac{\pi}{2}\right)[/math]
Using the graph of [math]f\left(x\right)=e^x[/math], which of the following statements are true about [math]f'\left(x\right)[/math]
Using the graph of [math]f\left(x\right)=\ln\left(x\right)[/math], which of the following statements are true about [math]f'\left(x\right)[/math]
Using the graph of [math]f\left(x\right)=\left|x\right|[/math], which of the following statements are true about [math]f'\left(0\right)[/math]
When the graph of [math]f[/math] is increasing, then [math]f'[/math] is
When the graph of [math]f[/math] is decreasing, then [math]f'[/math] is
When the graph of [math]f[/math] has a corner, then [math]f'[/math] is