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.
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 |
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.
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 |