Quinton Fischer Assignment Homework 07 F19 due 11/12/2019 at 07:00pm EST MA103 F19 4. (1 point) Find the particular antiderivative that satisfies the following conditions: 1. (1 point) Find the most general antiderivative F of f (x) = 3x2 + 1x + 8. √ F 0 (x) = x4 + 4 x; F(x) = F(1) = −1. F(x) = Answer(s) submitted: NOTE: Don’t forget the constant in your answer. • xˆ5/5+8xˆ(3/2)/3-58/15 Answer(s) submitted: (correct) • xˆ3+xˆ2/2+8x+c 5. (1 point) Find the particular antiderivative that satisfies the following conditions: (correct) 2. (1 point) Find the most general antiderivative F of H 0 (x) = f (x) = −6x2 − 3x + 10 . x 5 10 − ; x3 x5 H(1) = 0. H(x) = Answer(s) submitted: F(x) = • 5/(2xˆ4)-5/(2xˆ2) NOTE: Don’t forget the constant in your answer. (correct) Answer(s) submitted: 6. (1 point) Suppose f 00 (x) = − (sin(x)), f 0 (0) = −4, and f (0) = 5. • 10lnx-3xˆ2-3x+c (correct) 3. (1 point) Find the most general antiderivative y of Find f (π/2). f (π/2) = Answer(s) submitted: • 6-5pi/2 dy = 8ex + 3. dx (correct) Z 29 7. (1 point) Estimate I = y= 8 (7x − 7x2 )dx using a Riemann sum, midpoints, and n = 3 subintervals. NOTE: Don’t forget the constant in your answer. I≈ Answer(s) submitted: Answer(s) submitted: • 8eˆx+3x+C • -52393.25 (correct) (correct) 1 8. (1 point) Z 10 Estimate I = ∆x = (3x2 + 4x + 5) dx using a Riemann sum, 4 n = 3 subintervals, and xi = (a) Left endpoints. (b) Using the definition, evaluate the integral. I≈ (b) Right endpoints. Z 5 (4 + 3x) dx = −1 I≈ Answer(s) submitted: Answer(s) submitted: • 6/n • -1+6i/n • 60 • 870 • 1422 (correct) (correct) 9. (1 point) Z b 11. (1 point) Suppose that the integral I = Z π/2 Estimate I = f (x) dx can 0 x sin(x)dx 0 π π π π using the partition {0, , , , } and 6 4 3 2 be represented by the following Riemann sum using right endpoints and n subintervals: s (a) Left endpoints. s s 2 2 2 4 8 4n 4 4 4 16 − · + 16 − · + . . . + 16 − · n n n n n n I≈ (a) Determine b. b= (b) Right endpoints. (b) Determine f (x). I≈ f (x) = Answer(s) submitted: (c) Evaluate the integral by interpreting it in terms of areas. • 0.68878 • 1.342364 I= (correct) Answer(s) submitted: 10. (1 point) • 4 • sqrt(16-xˆ2) • 4pi Recall the definition of a definite integral: (correct) n Z b 12. (1 point) Consider the function f (x)dx = lim ∑ f (xi )∆x n→∞ a i=1 where xi are right endpoints. (a) Determine ∆x and xi in the definition for f (x) = Z 5 (4 + 3x) dx. −1 2 −5, if 1 ≤ x < 8 4, if 8 ≤ x ≤ 15 Evaluate the following definite integral using properties of definite integrals. Z 5 (b) 9 f (x) − 10 dx = 6 Answer(s) submitted: • -7 • 73 Z 15 f (x) dx = 1 (correct) 15. (1 point) The sum: Answer(s) submitted: • -7 Z 2 (correct) Z 5 f (x) dx + −2 13. (1 point) Given that 4 ≤ f (x) ≤ 6 for −6 ≤ x ≤ 1, use a property of definite integrals to find an interval in which the value of f (x) dx − can be written as a single integral in the form: f (x)dx lies. −6 ≤ Answer: Z b f (x) dx. f (x)dx ≤ a −6 Answer(s) submitted: • 28 • 42 Determine a and b. (correct) Z 7 Z 5 f (x) dx = 9, 14. (1 point) Suppose that 4 Z 7 a= f (x) dx = 10, 4 f (x) dx = 6. and b= 6 Answer(s) submitted: • -1 • 5 Evaluate the following definite integrals: Z 6 f (x) dx = (a) f (x) dx −2 2 Z 1 Z 1 Z −1 (correct) 5 Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 3