MATH 182 $\;$TEST I February 12, 1998 Name

SHOW ALL WORK Instructor/Section

1.

Determine whether the following integrals are convergent or divergent. For those integrals that are convergent, evaluate them.

2.

Evaluate the following integrals.

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3.

Express $${\lim _{n\rightarrow \infty} \; \sum_{i=1}^{n}\; \left( \frac{i}{n}\right) ^{10}\cdot \frac{1}{n}}\;\;$$ as an integral on the interval [0,1] .

4.

Approximate the integral ${\displaystyle \int_{0}^{2} x^{2}\;dx}\;$ by using a Riemann sum taking the sample points to be right-hand endpoints and n=4 .

5.

The left, right, trapezoidal, and midpoint rule approximations were used to estimate ${\displaystyle \int_{0}^{2} f(x)\;dx},$ where f(x) is the function whose graph is shown. The estimates were 6.1, 6.2, 6.3, and 6.4, and the same number of subintervals were used in each case. Fill in the blanks below.

(a)

The midpoint approximation is .

(b)

The trapezoidal approximation is .

(c)

The left endpoint approximation is .

(d)

The right endpoint approximation is .

6.

Suppose ${\displaystyle g(x)=\int_{-x^{2}}^{0} \cos t\;dt}.\;$ Find g'(x) dx .

7.

Let ${\displaystyle g(x)= \int_{0}^{x} f(t)\;dt},\;$ where f(t) is the function whose graph is shown below. Fill in the blanks.

(a)

Is g(4) positive or negative?

(b)

g(0)= .

(c)

Is the function g increasing or decreasing on the interval (5,6) ?

(d)

g has a local maximum, local minimum, or neither at x=10 .

(e)

The absolute maximum of g occurs at x= .

(f)

g has a local maximum, local minimum, or neither at x=7 .