Lab 17: The fundamental theorem of calculus - part 3


The following diagram represents eight different ways you might use the Fundamental Theorem of Calculus. For example, #7 represents using a table of values for a function to compute values for its antiderivative.

lab17 1

Classify each of the following problems as 1-8 from the diagram then solve the problem.

 

              Estimate\(\int_0^{100} f(t) dt\) given the following table.

 

\(t\) 0  20 40 60 80 100
 \(f(t)\)  1.2 2.8  4.0 4.7 5.1 5.2

 lab17 2

              Find \(F(-1.5)\) given the graph of \(f(x) = F'(x)\) and \(F(1) = 0\) shown to the right. 

 

              If \(f'(x) = e^{-x^2}\) and \(f(0) = 2\) find \(f(1)\) and \(f(-1)\).

 

              Find the area below the graph of \( y = \cos x^2\) from \(x = 0\) to \(x = \sqrt{\frac{\pi}{2} }\).

 

              Find \(\int_{-4}^4 f(x)dx\) given the graph shown to the right.

 lab17 3

              Find \(\int_0^1 3e^{-2x} dx\).

 

              Find the area between the graphs of \(y = x^2\) and \(y = 2x^2 + x - 6\).

 

              If \(f(0) = 100\) estimate \(f(x)\) for \(x =2,4,\) and \(6\) given the following table.

 

\(x\) 0  2 4 6
 \(f'(x)\)  10 18  23 25