| № |
Condition |
free/or 0.5$ |
| m52830 | (a) Use 4 to show that if sn is the nth partial sum of the harmonic series, then
Sn ≤ 1 + 1n n
(b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22. |
buy |
| m52831 | (a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation.
(b) Solve the differential equation.
(c) Use the CAS to draw several members of the family of solutions obtained in part (b). Compare with the curves from part (a).
y = y2 |
buy |
| m52832 | (a) Use a computer algebra system to find the partial fraction decomposition of the function
(b) Use part (a) to find ∫ f(x) dx (by hand) and compare with the result of using the CAS to integrate f directly. Comment on any discrepancy. |
buy |
| m52833 | (a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values.
(b) Estimate the value of x at which f increases most rapidly. Then find the exact value.
f(x) = x + 1/ √x2 + 1 |
buy |
| m52834 | (a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.
(b) Use a graph of f" to give better estimates.
F(x) = cos x + 1/2 cos 2x, 0 ≤ x ≤ 2π |
buy |
| m52835 | (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.
f(x) = x5 - x3 + 2, -1 ≤ x ≤ 1 |
buy |
| m52836 | (a) Use a graph to guess the value of the limit
(b) Use a graph of the sequence in part (a) to find the smallest values N of that correspond to ( = 0.1 and in Definition 2. |
buy |
| m52837 | (a) Use a graphing calculator or computer to graph the function f(x) = x4 - 3x3 - 6x2 + 7x + 30 in the viewing rectangle [-3, 5] by [-10, 50].
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of f .
(c) Calculate f (x) and use this expression, with a graphing device, to graph f . Compare with your sketch in part (b). |
buy |
| m52840 | (a) Use differ nation to find a power series representation for
What is the radius of convergence?
(b) Use part (a) to find a power series for
(c) Use part (b) to find a power series for |
buy |
| m52841 | (a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height h, inner radius r, and thickness Δr.
(b) What is the error involved in using the formula from part (a)? |
buy |
| m52842 | (a) Use Euler s method with each of the following step sizes to estimate the value of y(0.4), where y is the solution of the initial-value problem y = y, y(0) = 1.
(i) h = 0.4
(ii) h = 0.2
(iii) h = 0.1
(b) We know that the exact solution of the initial-value problem in part (a) is y = ex. Draw, as accurately as you can, the graph of y = ex, 0 < 0.4, together with the Euler approximations using the step sizes in part (a).
(Your sketches should resemble Figures 12, 13, and 14.)
Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates.
(c) The error in Euler s method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler s method to estimate the true value of y(0.4), namely e0.4. What happens to the error each time the step size is halved? |
buy |
| m52843 | (a) Use Formula 10.2.6 to show that the area of the surface generated by rotating the polar curve
r = f (θ), a ≤ θ ≤ b
(Where f is continuous and 0 ≤ a < b ≤ π) about the polar axis is
(b) Use the formula in part (a) to find the surface area generated by rotating the lemniscate r2 = cos 2θ about the polar axis. |
buy |
| m52844 | (a) Use the definition of a derivative to find f (2), where f(x) = x3 - 2x.
(b) Find an equation of the tangent line to the curve y = x3 - 2x at the point (2, 4).
(c) Illustrate part (b) by graphing the curve and the tangent line on the same screen. |
buy |
| m52846 | (a) Use the identity for tan(x - y) (see Equation 14b in Appendix D) to show that if two lines L1 and L2 intersect at an angle α, then
tan α = m2 - m1 / 1 + m1m2
where and are the slopes of and , respectively.
(b) The angle between the curves C1 and C2 at a point of intersection P is defined to be the angle between the tangent lines to C1 and C2 at P (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection.
(i) y = x2 and y = (x - 2)2
(ii) x2 - y2 = 3 and x2 - 4x + y2 + 3 = 0 |
buy |
| m52847 | a) Use the Midpoint Rule and the given data to estimate the value of the integral
b) If it is known that -2 ≤ f (x) ≤ 3 for all x, estimate the error involved in the approximation in part (a). |
buy |
| m52848 | (a) Use the Product Rule twice to prove that if , , and are differentiable, then (fgh) = fgh + fg h + gfh .
(b) Taking f = g = h in part (a), show that
d/dx[f(x)]3 = 3[f(x)]2 f (x)
(c) Use part (b) to differentiate y = e3x. |
buy |
| m52849 | (a) Use the Quotient Rule to differentiate the function
f(x) = tan x - 1/ sec x
(b) Simplify the expression for by writing it in terms of sin x and cos x, and then find f (x).
(c) Show that your answers to parts (a) and (b) are equivalent. |
buy |
| m52850 | (a) Use the reduction formula in Example 6 to show that
∫ sin2x dx = x/2 - sin 2x/4 + C
(b) Use part (a) and the reduction formula to evaluate
∫ sin4x dx |
buy |
| m52851 | (a) Use the reduction formula in Example 6 to show that
Where n ≥ 2 is an integer
(b) Use part (a) to evaluate |
buy |
| m52853 | (a) Use the sum of the first 10 terms to estimate the sum of the series
How good is this estimate?
(b) Improve this estimate using 3 with n = 10.
(c) Compare your estimate in part (b) with the exact value given in Exercise 34.
(d) Find a value of n that will ensure that the error in the approximate s ( sn is less then 0.001. |
buy |