Given the graph of [math]f\left(x\right)[/math] shown below. Graph [math]f'\left(x\right)[/math]and [math]f''\left(x\right)[/math] in the same window as the function. Then answer the questions below the graph.
The graph of the function, [math]f\left(x\right),[/math] is increasing when the graph of the derivative, [math]f'\left(x\right),[/math] is ________________ the [math]x[/math]-axis.
The graph of the function, [math]f\left(x\right),[/math] is decreasing when the graph of the derivative, [math]f'\left(x\right),[/math] is ________________ the [math]x[/math]-axis.
The graph of the function, [math]f\left(x\right),[/math] is concave down when the graph of the second derivative, [math]f''\left(x\right),[/math] is ________________ the [math]x[/math]-axis.
The graph of the function, [math]f\left(x\right),[/math] is concave up when the graph of the second derivative, [math]f''\left(x\right),[/math] is ________________ the [math]x[/math]-axis.
In the Geogebra window below the graph of [math]f\left(x\right)=4-x^2[/math] is shown. If we want to estimate the area under the curve to the right of the [math]y[/math]-axis, we can use Riemann Sums. You will notice in the graph that we are interested in the area between [math]x=0[/math] and [math]x=2[/math]. [br]To calculate the left-hand sum, use LeftSum([math]f(x)[/math], 0, 2, n), where [math]f(x)[/math] is the function and n is the number of rectangles.
Using the Input bar in the graph above, have Geogebra calculate the left-hand sum using 8 rectangles. Your answer is an ____________________ of the area.
Using the Input bar in the graph above, have Geogebra calculate the LowerSum using 8 rectangles. Your answer is an ____________________ of the area.
Now calculate the actual area using the Integral command. The actual area between the graph and the [math]x[/math]-axis is ______________ (round answer to 3 decimal places).
If we want to find the general antiderivative of a function, [math]f(x)[/math], we use the Integral command. A function is given in the graphing window below, use Integral([math]f(x)[/math]) to find the antiderivative.