Barrons AP Calculus
Example 13 __ . Example 14 __ . Example 15 __ by theorem (1). Example 16 __ by theorem (4), by theorem (1). E ...
. Example 19 __ Find . SOLUTION:. Here we have let and noted that where . The limit on the right in the s ...
C. DEFINITION OF DEFINITE INTEGRAL AS THE LIMIT OF A RIEMANN SUM Most applications of integration are bas ...
Write as the limit of a Riemann Sum. SOLUTION: The endpoints indicate that a = 1 and b = 4, so ...
What if f(x) is negative? Then any area above the graph and below the x-axis is counted as negative ( ...
Figure N6–3 Note that if f is continuous then the area between the graph of f on [a,b] and the x-axis ...
(a) (b) (c) (a) (b) E1. Using Rectangles We may approximate by any of the following sums, where Δx represents t ...
(c) approximating sum is . This sum uses the circumscribed rectangles shown in Figure N6–4. Figure N6–4 Figure ...
(d) . Since the exact value of or 4, the midpoint sum is the best of the three approximations. This is ...
(a) . Example 25 __ Use T(4) to approximate . SOLUTION: From Example 24, . Then, . This is better than eithe ...
(b) A ≈ 1(7) + 3(11) + 2(13) + 4(12) ≈ 114 In each subinterval, we sketch trapezoids by drawing ...
E3. Comparing Approximating Sums If f is an increasing function on [a,b], then , while if f is decrea ...
Figure N6–8 is helpful in showing how the area of a midpoint rectangle compares with that ...
Figure N6–9 SOLUTION: Since f increases on [a,b] and is concave up, the inequality is F. GRAPHING A FUNCTIO ...
Figure N6–10 SOLUTION: We know that if f ′ < 0 on an interval then f increases on the interv ...
Figure N6–11a Figure N6–11b All that remains is to evaluate f(x) at x = 1, 2, and 3. We use the Fundamenta ...
(a) (b) (c) (a) where the integral is the area of the triangle with height 1 and base 1; So the function ...
(b) (c) Figure N6–12a To calculate the distance traveled during the hour, we use the methods demo ...
Figure N6–12b Example 30 __ The graph of f(t) is given in Figure N6–13. If , fill in the values for F(x) ...
(i) (ii) (iii) Here is the completed table: Example 31 __ The graph of the function f(t) is shown in Figure ...
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