# Expert Answers

(1) (a) Use the Midpoint Rule with n = 4 subintervals to estimate

The width of each interval is:

(b) Use the Error Bound to find the bound for the error

When using the Midpoint Rule, the Error Bound is:

(c) Compute the integral exactly

(d) Verify the error is really no more than the error bound

(2) (a) Use Simpson’s Rule with n = 4 subintervals to estimate

(b) Use the Error Bound to find the bound for the error

(c) Compute the integral exactly

(d) Verify the error is really no more than the error bound

(3) Estimate using the Trapezoidal Rule with n = 6 subintervals

(4) Use the Error Bound formula for the Trapezoidal Rule to determine N so that if is approximated using the Trapezoidal Rule with N subintervals, the

error is guaranteed to be less than 10-4.

(1) Let y = g(x) be the inverse function of y = f(x) = x+ x + 1.

(a) Determine g(3).

(b) Determine g

(c) Determine the equation of the line tangent to y = g(x) at x = 3.

(2) Evaluate the de_nite integral

(3) Compute the derivative of f(x) =

1 + ln x

1 ln x

.

(4) Compute the de_nite integral

(5) Compute lim

(6) Use Integration by Parts to evaluate the de_nite integral

(7) Compute the inde_nite integral

(8) Compute the inde_nite integral

(9) Compute

(10) Determine the intervals where the functionf(x) = xe is increasing and the intervals where it is decreasing.

2. Be sure to evaluate the following integrals as improper integrals.

(1) Evaluate

(2) Evaluate . If you use a substitution to do the integration, be sure the change the limit of integration to the correct ones for the new variable.

(3) Evaluate

(4) Evaluate . You’ll need to use L’Hospital’s rule to calculate the limit that appears.

3. Generally speaking, arc length and surface area integrals are difficult to compute exactly since the integrands often do not have an antiderivative in terms of common functions. So approximation methods are used (such as the Trapezoidal Rule) to get an estimate of the integral. But, for all the problems below, the functions have been selected so that the integrals can be done exactly.

(1) Compute the length of the curve , for .

(2) Compute the length of the curve , for .

(3) Compute the length of the curve , for

(4) Compute the surface area generated by revolving the curve , for about the x-axis.

(1) (a) Eliminate the parameter from

(b) Sketch the graph of that curve for parameter values .

Indicate the implied direction the curve is traced out.

To determine the implied direction that the curve is traced out, make a table of values for x and y for selected values of t in the domain:

(c) Find the equation of the line tangent to the curve at t = 2.

(2) (a) Eliminate the parameter from

(b) Sketch the graph of that curve for parameter values

Indicate the implied direction the curve is traced out.

(c) Find the equation of the line tangent to the curve at .

(c) Find the equation of the line tangent to the curve at .

(3) Let C be the curve given parametrically by

(a) Find an equation for the line tangent to C at the point corresponding to .

(b) Determine the values of t where the tangent line is horizontal or vertical.

(4) Compute the length of the curve given by

The a is a positive constant.

The width of each interval is:

(b) Use the Error Bound to find the bound for the error

When using the Midpoint Rule, the Error Bound is:

(c) Compute the integral exactly

(d) Verify the error is really no more than the error bound

(2) (a) Use Simpson’s Rule with n = 4 subintervals to estimate

(b) Use the Error Bound to find the bound for the error

(c) Compute the integral exactly

(d) Verify the error is really no more than the error bound

(3) Estimate using the Trapezoidal Rule with n = 6 subintervals

(4) Use the Error Bound formula for the Trapezoidal Rule to determine N so that if is approximated using the Trapezoidal Rule with N subintervals, the

error is guaranteed to be less than 10-4.

(1) Let y = g(x) be the inverse function of y = f(x) = x+ x + 1.

(a) Determine g(3).

(b) Determine g

(c) Determine the equation of the line tangent to y = g(x) at x = 3.

(2) Evaluate the de_nite integral

(3) Compute the derivative of f(x) =

1 + ln x

1 ln x

.

(4) Compute the de_nite integral

(5) Compute lim

(6) Use Integration by Parts to evaluate the de_nite integral

(7) Compute the inde_nite integral

(8) Compute the inde_nite integral

(9) Compute

(10) Determine the intervals where the functionf(x) = xe is increasing and the intervals where it is decreasing.

2. Be sure to evaluate the following integrals as improper integrals.

(1) Evaluate

(2) Evaluate . If you use a substitution to do the integration, be sure the change the limit of integration to the correct ones for the new variable.

(3) Evaluate

(4) Evaluate . You’ll need to use L’Hospital’s rule to calculate the limit that appears.

3. Generally speaking, arc length and surface area integrals are difficult to compute exactly since the integrands often do not have an antiderivative in terms of common functions. So approximation methods are used (such as the Trapezoidal Rule) to get an estimate of the integral. But, for all the problems below, the functions have been selected so that the integrals can be done exactly.

(1) Compute the length of the curve , for .

(2) Compute the length of the curve , for .

(3) Compute the length of the curve , for

(4) Compute the surface area generated by revolving the curve , for about the x-axis.

(1) (a) Eliminate the parameter from

(b) Sketch the graph of that curve for parameter values .

Indicate the implied direction the curve is traced out.

To determine the implied direction that the curve is traced out, make a table of values for x and y for selected values of t in the domain:

(c) Find the equation of the line tangent to the curve at t = 2.

(2) (a) Eliminate the parameter from

(b) Sketch the graph of that curve for parameter values

Indicate the implied direction the curve is traced out.

(c) Find the equation of the line tangent to the curve at .

(c) Find the equation of the line tangent to the curve at .

(3) Let C be the curve given parametrically by

(a) Find an equation for the line tangent to C at the point corresponding to .

(b) Determine the values of t where the tangent line is horizontal or vertical.

(4) Compute the length of the curve given by

The a is a positive constant.

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