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restart; |

> |

Maple 3

**The Definite Integral** and Differential Equations.

Open a maple worksheet and answer the questions below.

See http://www.math.uri.edu/Center/workht/calc1/integral1.html

and

http://www.math.uri.edu/Center/workht/calc1/diffeq1.html

for helpful information.

**Homework Problems**

**Problem 1. ** Find exact values and a floating point approximations of the following integrals:

(a) .

(b)

**Problem 2. **The amount of a certain drug in the bloodstream following an injection changes at the rate

in mg per hour, where t is the time, in hours, after the injection. Assume that there was 30 mg of the drug in the bloodstream at the moment of injection.

(a) Find a function d(t) which gives the amount of the drug in the bloodstream at time t.

(b) Plot d(t) and c(t) in one coordinate system for the twelve hour period following the injection.

(c) At what point does the amount of the drug reach its maximum?

(d) What is the maximal amount of the drug?

Problem 3:

**Remark. **As you can guess, Maple can easily calculate left and right Riemann sums and visualize them as areas of rectangles. The appropriate commands "leftsum", "leftbox" , "rightsum", "rightbox" are contained in the package "with(student)", which, similarly as the package "with(plots)", has to be loaded into the computer memory before you can use it. You load the package with the command: "with(student):", if you do not want Maple to print the content of the package, or "with(student);" , if you do. Press enter on the command lines below and see what happens.

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f :=x->x^2; |

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with(student); |

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leftbox(f(x),x=0..1,50); |

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rightbox(f(x),x=0..1,50); |

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leftsum(f(x),x=0..1,50); evalf(%); |

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rightsum(f(x),x=0..1,50); evalf(%); |

Problem 4 Use dsolve to find the solution of the differential equation y' = x + y.

Problem 5 Use dsolve to find the solution of the initial-value problem y' = , y (1) = -5.