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Written assignment #4

Math& 148

Write careful, complete solutions for the following problems and submit them according to Directions for Submitting Written Assignments, under Orientation to Online MATH 148 in the Canvas classroom.

Note

The named functions we use in this class are so-called elementary functions. This name is attached to functions familiar to mathematicians in the 27th and 18th Century, such as powers, roots, exponentials logarithms, trigonometric functions (which we don’t consider in this class), and combinations of these, via arithmetical operations and composition.

It turns out that derivatives of elementary functions are also elementary functions. It may take many steps, but that’s why no matter how complicated an expression you may find, we can always write an explicit derivative function of it. Integration is quite different: in a sense that can be made precise, most elementary functions do not have an elementary antiderivative. That means that for many important problems, we have to rely on numerical integration (a

typical example is the function e− x 2

2 , which is basic in many statistics problems, and has no elementary antiderivative, so that to work with its integrals we rely on software or tables).

1

The function ex+2 ln x does not have an elementary antiderivative. Try to fins a technological tool (Wolfram Alpha, more on line sites, numerical software for your computer, a powerful enough calculator...) to estimate

ˆ 4

1

ln x · ex+2dx

Tell me what you used and how you used it.

Note: To find my own estimate, I relied on wxMaxima, a free, open source software package, based on a mainframe software developed many years ago, called Maxima.

1

2 2

2

The table shows values of the function y = G(t). Use he table to estimate the

value of ´50

0 G(t)dt

t 0 10 20 30 40 50

G(t) 15 19 8 −12 −3 15 Of course, lacking more detail, we would not be able to give a reliable esti-

mate of the error (e.g., we can’t evaluate an upper and a lower integral).

3

Find at least two different antiderivatives of f(t) = t3 + et

4

Use antiderivatives to compute the following definite integral (do not use nu- merical methods, software or otherwise)

ˆ

9

4

(

√

t − 1

3t +

5

t4

)

dt

5

The graph shown here is the graph of the function g′(t)

Knowing that g(0) = 5, use the graph to find the value of

1. g(2)

2. g(5)

6 Ch.3#24 3

6 Ch.3#24

7

Represent with a definite integral the area bounded by the curve y = √ x, the

x−axis, and the line x = 9, and give its value

8 Ch.3#82