Math 121: Chapter 4 Exam Study Guide

1. Find the indefinite integral; show work. Five parts.
2. Evaluate the definite integrals. Show work and give exact answers. Three parts.
3. Write the expression in summation notation. Example \(\frac 1 2 + \frac 2 3 + \frac 3 4 + \frac 4 5 + \cdots + \frac {19}{20} = \sum\limits_{k = 1}^{19} {\frac {k}{k+1}} \) . You do not need to evaluate the sum, just write it in summation notation. Two parts.
4. You are given the closed forms for the sums of one, the sum of the integers, and the sum of the squares of the integers. Use it to find a sum for a value of n, the general closed form, and the limit as n approaches infinity of an expression.
5. Given the values of two definite integrals, use them to find other definite integrals. Five parts.
6. Evaluate the limit of a sum by first writing it as a definite integral. For example, \(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left( {{{\left( {9 + {\textstyle{{4k} \over n}}} \right)}^2} + 3\left( {9 + {\textstyle{{4k} \over n}}} \right) - 2} \right)\left( {\frac{4}{n}} \right) = \int_9^{13} {\left( {{x^2} + 3x - 2} \right)dx} = \frac{{1840}}{3}} \).
7. A region is shown. Find the area using geometric formulas. Then write the area as the sum of definite integrals. Do not evaluate the integrals.
8. Given a function where the upper limit is defined in terms of a variable/function, find the indicated values. Example: Let \(F(x)=\int_0^x \sin t\, dt\) . Find \(F(0)\), \(F(\pi)\), \(F'(0)\), \(F'(\pi)\), \(F''(0)\), and \(F''(\pi)\)
9. Use the second fundamental theorem of calculus to find the derivative. Two parts.
10. Rewrite the integral and limits in terms of u. Do not evaluate the integral after rewriting. Two parts.
11. Numerical integration -- take home.
12. Numerical integration -- take home.

Notes

• Most of the problems are similar to those in the textbook.
• Don't spend too long on any one problem. If you get stuck, make a note to come back and then move on.
• You can verify the answer in #6 with Maxima using this command: limit(nusum(((9+4*k/n)^2+3*(9+4*k/n)-2)*(4/n),k,1,n),n,inf); See the Calculus Wiki for an explanation and another example.
• There is a take home exam worth 20 points that is due the day of the in-class test.

Points per problem

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Last updated November 2, 2011 11:54 PM