Calculus Expert Answers & Study Resources

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) What is the least squares error E for the points (1, -3), (-2, 5), and (0, 10) and the line y = Ax + B? A) E = (A + B + 3)2 + (-2A +.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Let f(x, y) = 7x + 2y - 7. Compute f(-7, -2). A) -67 B) -53 C) 12 D) -62 2) Let g(x, y) = 6y2 - 2xy. Compute g(7, -2). A) 320 B) 45 C) 52 D) 322 3) Let g(x, y) = (x.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. 2ty' - y = (6/t); y(1) = -1, t > 0 A) integrating factor: 2t general solution: y = - (3/2t2) + (C/2t) particular.

Find the second-order partial derivative. 19) Let H(x, y) = (3xy/x2 - y). Find (∂2H/∂y2). A) (9x4 - 6x3 - 9x2y/(x2 - y)3) B) (3x3/(x2 - y)2) C) (6x3/(x2 - y)3) D) (6x3y - 6x5/(x2 - y)4) E) none of these 20) Let F(x, y, z) = (xz/y2z + x) - 5x2y3. Compute (∂F/∂y)(-1, 0, 1). A) -.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 25) Let f(x) = (1/x + 1). Determine the second Taylor polynomial p2(x) of f(x) at x = 0. A) 1 - x B) 1 - 2x + 2x2 C) 1 - x + x2 D) 1 - x -.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative of the function. 1) y = 8cos10x A) (dy/dx) = 80sin10x B) (dy/dx) = -80sin10x C) (dy/dx) = -8sin10x D) (dy/dx) = 10sin10x 2) y = 5sin(5x - 5) A) (dy/dx) = 5cos(5x - 5) B) (dy/dx) = -5cos(5x - 5) C).

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 35) Find the pairs (x, y) that give the extreme values of 2x + 10y, subject to the constraint 4x2 + 5y2 = 8400, using the method of Lagrange multipliers. Enter your answer as exactly just.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 45) Consider . Which of the following is true? A) The limit diverges. B) The limit exists and is equal to zero. C) none of these D) The limit exists and is equal to two. E) The limit exists and is.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. For the given probability density function, over the stated interval, find the requested value. 1) f(x) = (1/7), over the interval 2 ≤ x ≤ 8. Find E(x). A) (60/7) B) (59/14) C) 24 D) (30/7) 2) f(x) = (1/5)x, over the.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 21) Use two repetitions of the Newton-Raphson algorithm to find the value of x near zero for which cosx = x. Enter just a real number rounded off to two decimal places. 22) Use two repetitions of.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 26) Find an infinite series that converges to the value of dx. Is (1/3) - (1/6) + (1/18) - (1/72) ... correct? Enter "yes" or "no". 27) Find an infinite series that converges to the value of.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 35) Find the value of k that makes f(x) = kx2 a probability density function on the interval 0 ≤ x ≤ 1. Enter just an integer. 36) Find the value of k that makes f(x) =.

Solve the differential equation with the given initial condition. 23) y' = y + 1, y(0) = 1 A) y = 2et -1 B) y = t2 + 1 C) y = t + 1 D) y = et - 2 24) y' = ty, y(0) = -1 A) y = -et^2/2 B) y = -1 + et^2/2 C).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Newton-Raphson algorithm to approximate the given root to the nearest thousandth. 1) √(2) A) 1.419 B) 2.000 C) 1.411 D) 1.414 2) (3)√(4) A) 1.578 B) 1.593 C) 1.583 D) 1.587 Use the Newton-Raphson algorithm to find a zero of the function on the given.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 61) ∫(1/xlnxln(lnx))dx [Hint: Let u = ln(lnx).] Enter your answer with any coefficients in front as integers or reduced fractions of form (a/b). 62) ∫(sec2(lnx)/x)dx Enter your answer with any coefficients in front as integers or reduced fractions.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Convert 327° to radian measure. A) (109/120) B) (109/60) C) (109π/120) D) (109π/60) 2) Convert -150° to radian measure. A) (7/6) B) - (5/3)π C) (5/3) D) - (5/6)π E) none of these SHORT ANSWER. Write the word or phrase that best completes each statement or.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 19) sinxcos3x Enter cosine terms first, no parentheses around arguments. 20) sinx2 Do not enter parentheses around arguments. 21) 3cosxsin2x Enter cosine terms first, no parentheses around arguments. 22) cos2(x + 4) Enter cosines before sines with parentheses around arguments. 23) sin3(x3) Enter sine.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A set of exam scores is 80, 75, 85, 90, 100, 70, 60. The standard deviation equals A) √(50) B) √(20) C) √(10) D) 7 E) none of these 2) The table below is the probability table for a random variable.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the partial derivative. 1) f(x, y) = 10x - 10y2 - 4. Find (∂f/∂x). A) 6 B) 10 C) 10x D) -20y 2) f(x,y) = 3x2 - 15xy + 7y3. Find (∂f/∂x). A) -15x + 21y2 B) 6x2 - 15x C) 6x - 15y D).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the integral by making an appropriate substitution. 1) ∫((x7 - 2x6)5(7x6 - 12x5))dx A) (1/5)(x7 - 2x6)5 + C B) (1/6)(x7 - 2x6)6 + C C) 7x6 - 12x5 + C D) (x7 - 2x6)6 + C 2) ∫((6x5dx/(4 + x6)3)) A).

Solve the initial value problem. 21) y' + ty = 2t; y(0) = 5 A) y = 2e-t^2/2 + 3 B) y = 3et^2/2 + 2 C) y = 2et^2/2 + 3 D) y = 3e-t^2/2 + 2 22) 2y'- 4ty = 8t; y(0) = 11 A) y = -2 + 13e-t^2 B) y = -2 + 13et^2 C).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Let f(x, y) = x2 + 2xy + 5y2 + 2x + 10y - 3. At which point(s) does f(x, y) have possible maximum/minimum values? A) (-1, 17) and (0, 5) B) (1, 0) C) (-1, 0) and.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Divide the given interval into n subintervals and list the value of Δx and the endpoints a0, a1, ... , an of the subintervals. 1) -5 ≤ x ≤ 2; n = 5 A) Δx = 1.4; a0.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) Find a constant solution of y' = t(y - 1). Enter just an integer. 2) Find a constant solution of y' = 10 y - 7. Enter just a reduced fraction of form (a/b). MULTIPLE CHOICE. Choose the.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1) dx A) (12/35) B) 2 C) (1/105) D) (109/9) 2) dt A) (63/2) B) (75/2) C) (1827/4) D) (87/2) 3) dx A) (72/145) B) (8088/88,410,125) C) (8/145) D) (42/145) 4) dx A) (52√(26) + 250/3) B) 2√(26) -10 C) (52√(26) - 250/3) D) (52√(26) - 250/12) 5) dx A) 3e4 - 1 B) 3e4 C) 3e4.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the third Taylor polynomial of the function at x = 0. 1) f(x) = (1/x + 3) A) (1/3)x - (1/9)x2 + (1/27)x3 - (1/81)x4 B) (1/3)x + (1/9)x2 + (1/27)x3 + (1/81)x4 C) (1/3) - (1/9)x + (1/27)x2.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral using integration by parts. 1) ∫(4xexdx) A) 4ex - 4xex + C B) 4xex - 4ex + C C) 4ex - ex + C D) xex - 4ex + C 2) ∫(5xlnxdx) A) (5/2)xlnx - (5/4)x + C B) (5/2)x2lnx -.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 41) Determine the sum of the following geometric series: (22/53) + (24/55) + (26/57) + (28/59) + (210/511) + ... . Enter just a reduced fraction of form (a/b). 42) Determine the sum of the following geometric.

Determine whether the function is a probability density function over the given interval. 18) f(x) = (1/2), 4 ≤ x ≤ 10 A) Yes B) No 19) f(x) = 5x, 0 ≤ x ≤ 1 A) No B) Yes 20) f(x) = (1/2)x, 0 ≤ x ≤ 2 A) No B) Yes 21) f(x) = (1/4)x, 4 ≤ x ≤.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 23) Approximate dx; n = 4, by (a) Simpson's rule and (b) the trapezoidal rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma. 24) Approximate.

Find the producer surplus for the supply curve at the given sales level, x. 61) p = 2x2; x = 1 A) $0.67 B) $4 C) $1.33 D) $1 62) p = x2; x = 4 A) $42.67 B) $12 C) $64 D) $2.67 63) p = x2 + 1; x = 1 A) $0.67 B) -$0.33 C) $1 D) -$1 For a particular commodity, the.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 21) f(x) = (sinx2/tanx) Enter a quotient with cosine and sine functions last in their terms. No parentheses or labels or multiplication dots. 22) f(x) = ln(tanx) + tanx Enter your answer exactly in the form a⋅(b +.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Integral Test to determine whether the series converges. 1) A) Diverges B) Converges 2) A) Converges B) Diverges SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) Use the integral test to.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Give the value of sint where t is the radian measure of the angle shown. A) 2 B) -2 C) (2/√(8)) = (√(2)/2) D) - (2/2) = -1 E) none of these SHORT ANSWER. Write the word or phrase that best.

Evaluate the integral using integration by parts. 21) ∫(x)√(x + 1)dx A) (2/3)x(x + 1)3/2 - (4/15)(x + 1)5/2 + C B) x(x + 1)3/2 - (x + 1)5/2 + C C) (2/3)(x + 1)3/2 - (4/15)(x + 1)5/2 + C D) (x + 1)3/2 + (x + 1)5/2 + C 22) ∫(eaxcosbxdx) A) eax(sinbx + cosbx).

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 21) Find (by inspection) the expected value and standard deviation of the random variable with the following density function: f(x) = (7/√(2π))e-49/2(x - 3.9)^2 Enter your answer as just two numbers (a real number to one.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Suppose (π/4) ≤ x ≤ (3π/4). Which of the following statements can be made abouttanx? A) tanx is positive. B) tanx is not defined. C) Iftanx is defined, then |tanx| ≥ 1. D) tanx is defined. E) none of these SHORT.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false: 1) f'(0) = 2 A) True B).

Determine the integral by making an appropriate substitution. 21) ∫((2x/√(x - 4)))dx A) 3(x - 4)3/2 + C B) (4/3)√(x + 4)(x - 8) + C C) (4/3)(x - 4)3/2 + C D) (4/3)√(x - 4)(x + 8) + C 22) ∫(tan2xsec2xdx) Use the substitution u = tanx. A) (tan2x/2) + C B) (sec2x/2) + C C) (tan3x/3) +.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 55) ∫(sintcost dt) Enter your answer in terms of cosine without using parentheses. 56) ∫(sin(3t + 2)dt) Enter your answer using parentheses around the argument of the function. 57) ∫(sin4t dt) Enter your answer without parentheses around the argument of.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 52) A random variable X has a probability density function f(x) = (x/32), 0 ≤ x ≤ 8. Find a such that Pr(X ≥ a) = (1/4). Enter your answer exactly in the reduced form b√(c),.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 37) Find the tangent line to the graph of f(x) = sinx + cosx at (π, -1). A) y = -x + 1 B) y = x + π C) y = π D) y = -x + π -1 E).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 18) Find all points (x, y) where f(x, y) = e(x^2 + y^2) has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of.

Consider the differential equation y' = g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false. 40) y = -3, y = 1, and y = 5 are the constant solutions to y' = g(y). A) True B) False 41) y = 2 is the.

Provide an appropriate response. 41) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 14.14 onces and a standard deviation of 0.04 ounce. Find the.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Enter your answer with any coefficients in front as integers or reduced fractions of form (a/b). 41) ∫(ex + xex/4 + xex)dx Enter your answer with any coefficients in front as integers or reduced fractions of form.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) The logistic differential equation (dP/dt) = 0.09P(800 - P) describes the growth of a population P, where t is measured in years. Find the carrying capacity of the population. A) 7.2 B) 400 C) 800 D) 1600 2) The logistic differential.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the general solution for the differential equation. 1) y(dy/dx) = x2 - 2x A) y2 = (2/3)x3 - 2x2 + C B) y2 = (1/3)x3 - 2x2 + C C) y = (1/3)x3 - 2x + C D) y =.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 39) Does this integral ∫(lnxdx) = xlnx + x + C? Enter "yes" or "no". 40) Does this integral ∫(excosxdx) = (exsinx + excosx/2) + C? Enter "yes" or "no". 41) Does this integral ∫(3xcosxdx) = 3xsinx + 3cosx.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 38) Let f(x, y) = (x2y3 + x3)5. Find (∂f/∂y). Is 5(x3 + y3x2)4(3y2x2) the correct answer? Enter "yes" or "no". 39) Let f(x, y) = x2exy. Find (∂f/∂x). Is (yx2 - 2x)exy the correct answer? Enter "yes".