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Chapter 4: Q. 5 (page 372)
What important theorem is the key to proving the Fundamental Theorem of Calculus?
The key to proving the Fundamental Theorem of Calculus is the Mean Value Theorem.
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Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating .
Explain why it would be difficult to write the sum in sigma notation.
Show by exhibiting a counterexample that, in general, . In other words, find two functions f and g such that the integral of their quotient is not equal to the quotient of their integrals.
For each function f and interval [a, b] in Exercises 27鈥�33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.
, n = 3 with
a) Trapezoid sim b) Upper sum
Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: ).
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