Use l'Hôpital's rule to find the limit. limx→0 6x2/cos x - 1 What must be done to put the given expression in a form to which l'Hôpital's rule can be directly applied? A. Nothing, the given expression already has the indeterminate form 0/0 . B. Nothing, the given expression already has the indeterminate form ∞/∞ . C. Nothing, the given expression already has the indeterminate form 0⋅∞. D. The expression must be rewritten as limx→06x2 (sec x - 1). Find the limit. limx→0 6x2/cos x-1 = (Type an exact answer.)
Find to two decimal places, the area of the surfaces generated by revolving the curve y = 4sinx in 0 ≤ x ≤ π about the x-axis. The area of the surface is (Simplify your answer. Type an integer or decimal rounded to two decimal places as needed.)
The fuel tanks for airplanes are in the wings, cross section below. The tank must hold 5400 lb of fuel with density 42 lb/ft3. Estimate the length of the tank using Simpson's Rule. Horizontal spacing = 1.6 ft. y0 = 1.4 ft, y1 = 1.5 ft, y2 = 1.7 ft, y3 = 2 ft, y4 = 2.2 ft, y5 = 2.4 ft, y6 = 2.1 ft The length of the tank is ft. (Round to the nearest tenth as needed.)
Use l'Hôpital's Rule to find the following limit. lim(0 - x)cscx x → 0- What must be done to put the given expression in a form to which l'Hôpital's rule can be directly applied? A. Nothing, the given expression already has the indeterminate form ∞/∞ . B. Nothing, the given expression already has the indeterminate form 0/0. C. The expression must be rewritten as limx→0 - (0-x)/sinx D. The expression must be rewritten in the expanded form. Find the limit. lim(0 - x)/cscx = (Type an exact answer.) x → 0-
The instructions for the given integral have two parts, one for the trapezoidal rule and one for Simpson's rule. Complete the following parts. ∫ 1 -1 (3x^2 + 7)dx. Using the trapezoidal rule a. Estimate the integral with n = 4 steps and find an upper bound for |ET|. T = (Type an exact answer. Type an integer or a simplified fraction.) An upper bound for |ET| is (Round to two decimal places as needed.) b. Evaluate the integral directly and find |ET|. ∫-1 1 (3x^2 + 7)dx = (Type an exact answer. Type an integer or a simplified fraction.) |ET| = (Round to two decimal places as needed.) c. Use the formula (|ET|/( true value )) × 100 to express |ET | as a percentage of the integral's true value. % (Round to the nearest integer as needed.) II. Using Simpson's rule a. Estimate the integral with n = 4 steps and find an upper bound for |ES|. S = (Type an exact answer. Type an integer or a simplified fraction.) An upper bound for |ES| is b. Evaluate the integral directly and find |ES |. ∫-1 1 (3x^2 + 7)dx = (Type an exact answer. Type an integer or a simplified fraction.) |ES| = c. Use the formula (|ES |/( true value) ) × 100 to express |ES| as a percentage of the integral's true value. % (Round to the nearest integer as needed.)
A rectangular swimming pool is 38 ft wide by 75 ft long. The table gives depths (d) from x = 0 at the shallow end to the diving end. Use the Trapezoidal Rule with n = 15 to estimate the volume of the pool, V = ∫0 75 38⋅d(x)dx. The volume of the pool is ft3. (Round to the nearest integer as needed.)
Find the following limit. lim x→0+ (x)6x What must be done to put the given expression in a form to which l'Hôpital's rule can be directly applied? A. Nothing, the given expression already has the indeterminate form 0/0. B. The expression must be rewritten as limx→0+ 1/(x)-6x. C. Nothing, the given expression already has the indeterminate form ∞/∞. D. The expression must be rewritten by taking the natural logarithm. Find the limit. lim x → 0+ (x) 6x = (Type an exact answer.)
The length of an ellipse with axes of length 2a and 2b is given below, where e is the ellipse's eccentricity. The integral in this formula, called an elliptic integral, is nonelementary except when e = 1 or 0. Complete parts (a) and (b) below. Length = 4a 0 π/2 √1 - e2cos2tdt, e = √a2 - b2/a a. Use the Trapezoidal Rule with n = 10 to estimate the length of the ellipse when a = 8 and e = 1/2. (Round to three decimal places as needed.) b. Use the fact that the absolute value of the second derivative of f(t) = √1 - e2cos2t is less than 1 to find an upper bound for the error in the estimate you obtained in part (a). |ET| ≤ (Round to four decimal places as needed.)
Find the centroid of the infinite region in the first quadrant between the curve y = e-x and the x-axis. (x, y) = (Type an ordered pair. Simplify your answer.)
Use l'Hôpital's Rule to evaluate the limit. lim x → 0 + (csc(4x) - cot(4x) + cos x) What must be done to put the given expression in a form to which l'Hôpital's rule can be directly applied? A. Nothing, the given expression already has the indeterminate form 0⋅∞. B. Nothing, the given expression already has the indeterminate form ∞/∞. C. The expression must be rewritten as limx→0 + 1-cos(4x)/sin(4x) + cos x. D. Nothing, the given expression already has the indeterminate form 0/0. Find the limit. lim x → 0 + (csc(4x) - cot(4x) + cos x) = (Type an exact answer.)
The sine-integral function, shown below, is one of the many functions in engineering whose formulas cannot be simplified. There is no elementary formula for the antiderivative of (sin t)/t. Complete parts (a) through (c) below. Si(x) = ∫x 0 sint/t dt 1 Click the icon to view more information. a. Use the fact that |f^(4)| ≤ 1 on [0, π/2] to give an upper bound for the error that will occur if Si (π/2) = ∫0 π/2 sint/t dt is estimated by Simpson's Rule with n = 4. (Round to five decimal places as needed.) b. Estimate Si (π/2) by Simpson's Rule with n = 4. (Round to five decimal places as needed.) c. Express the error bound you found in part (a) as a percentage of the value you found in part (b). % (Round to three decimal places as needed.)
Perform a test to determine if ∫-∞ ∞ dx √x^2+9 converges or diverges. Does ∫-∞ ∞ dx √x^2+9 converge or diverge? Converge Diverge
Estimate the minimum number of subintervals to approximate the value of ∫0 3 1/√5x+3 dx with an error of magnitude less than 10^-4 using a. the error estimate formula for the Trapezoidal Rule. b. the error estimate formula for Simpson's Rule. The minimum number of subintervals using the Trapezoidal Rule is (Round up to the nearest whole number.) The minimum number of subintervals using Simpson's Rule is (Round up to the nearest even whole number.)
Evaluate ∫1 9 7/s^2 ds using the trapezoidal rule and Simpson's rule. Determine i. the value of the integral directly. ii. the trapezoidal rule estimate for n = 4. iii. an upper bound for |ET|. iv. the upper bound for |ET| as a percentage of the integral's true value. v. the Simpson's rule estimate for n = 4. vi. an upper bound for |ES|. vii. the upper bound for |ES| as a percentage of the integral's true value. The value of ∫1 s 9 7/s^2 ds is (Round to four decimal places as needed.) The trapezoidal rule estimate of ∫1 9 7/s^2 ds for n = 4 is (Round to four decimal places as needed.) The upper bound on |ET | is (Round to four decimal places as needed.) The upper bound for |ET| as a percentage of the integral's true value is (Round to four decimal places as needed.) The Simpson's rule estimate of ∫1 9 7/s^2 ds for n = 4 is (Round to four decimal places as needed.) The upper bound on |ES| is (Round to four decimal places as needed.) The upper bound for |ES| as a percentage of the integral's true value is (Round to four decimal places as needed.)
Perform a test to detemine if ∫0 ln 3 3x -2e -3/xdx converges. ∫0 ln 3 3x -2e -3/xdx = (Type an exact answer.) Based on the test, does ∫0 ln3 3x -2e -3/xdx converge or diverge? Diverge Converge
Estimate the minimum number of subintervals to approximate the value of ∫-4 4 (4x^2 + 6)dx with an error of magnitude less than 3×10-4 using a. the error estimate formula for the Trapezoidal Rule. b. the error estimate formula for Simpson's Rule. The minimum number of subintervals using the Trapezoidal Rule is (Round up to the nearest whole number.) The minimum number of subintervals using Simpson's Rule is (Round up to the nearest even whole number.)
Find the area of the infinite region in the first quadrant between the curve y = e^-x and the x-axis. The area of the infinite region is. (Type an integer or a simplified fraction.)