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2. Let g(x)=x-2√x. 3 2 (a) Use GeoGebra to compute 1 (b) Below is the graph of y g(x). Use GeoGebra to find the total positive area of the shaded regions (as if you were going to cover them with carpet and needed to know positive areas for both regions). Write down the integral(s) you computed. 3 2 1 -1 0 1 2 3 4 5 3. Let f(x) = x - x³ - 2x² + 5x + 1 and g(x) = 2x - x7. Use GeoGebra to find the exact area (a fraction in lowest terms) of the shaded region in the graph below (just what you see in the window, nothing else). Write down the integral(s) you computed to find your answer. y = f(x) Sº 6 (c) You should get different answers to parts (a) and (b)! Explain why. 0.5 y = g(x) g(x) dx and write down your answer. 7 8 9
4. For some basic functions (like arctan ar, Inz, and tan r) we can't find an antiderivative just by invert- ing a derivative formula we already know. These examples do have antiderivatives, however - just not "nice" ones. Have GeoGebra compute this indefinite integral, and write the answer: farct arctanr dra Notice that GeoGebra adds an arbitrary con- stant to the end of antiderivatives (like c₁, or O, etc.) and makes a slider for the con- stant. GeoGebra also sketches the graph of the antiderivative, and as you use the slider to change the constant, GeoGebra changes the graph. At right, sketch a graph of the an- tiderivatives of arctan x that go with the con- stants 0, 2 and -3. 5. The function f(x) = e-² is an important function in many applications of calculus, but it is also an example of a simple function whose antiderivative cannot be made from algebraic, trigonometric, exponential or logarithmic operations. (a) Have GeoGebra compute fe-² dx. Write down the answer you get: (b) A numerical approximation to an integral like fe- dx is important in statistics. Have GeoGe- bra compute it and round to four decimal places:
6. For this problem let f(x) = √sin 2. (a) Sketch a graph of f(x) from a = 0 to = 16. It's kind of a weird one! Label the x-intercepts on the graph. (b) Now have GeoGebra find the antiderivative of f(x). What is GeoGebra's response? (c) Instead, have GeoGebra compute f(x) dx. State the result: GeoGebra should have given you a decimal approximation, since it can't figure out an an- tiderivative for f(x) that it can use in the FTC. Notice that there's no option in GeoGebra's output to turn the result into a simplified fraction with radicals, for example. You can force GeoGebra to find an approximation to a definite integral with the NIntegral command. This command approximates a definite integral using some type of Riemann sum. 1 (d) Use NIntegral to compute So sin(x²) dr. If you just use Integral the result will look bonkers. Write down the decimal result instead: 7. Suppose that v(t) = t+ecost is the velocity of a moving object at time t, measured in feet per second. Find a decimal approximation to the distance traveled by this object from t = 0 to t = 10. Write out the GeoGebra command that you use and put the correct unit on your answer. 8. GeoGebra is able to find the exact value of √²-4 da. First use Integral (the answer is really messy!), and then tell GeoGebra to show the decimal approximation - round to the nearest thousandth. Write down both answers. 11