Lab Final: MAT 229, Spring 2025

Name:

You may not skip problem 1, but are to select 6 of the remaining 9 problems to do (skip 3). But now you have Mathematica to give you a hand! I would suggest you choose the same problems as yesterday, but will permit you to swap out any. Note that there are some slight changes to certain problems. Good luck! When you are done, email this file to longa@nku.edu.

(20 pts) Consider the function f(x)=x ln(x).

Compute the limit: (lim_(x->(SuperPlus[0]))) f(x)

Compute the derivative of f(x), and find the value of x at which it equals zero.

Compute the integral: ∫ f(x) d x

Compute the integral numerically, using trapezoidal and two subintervals: (∫_1^2)f(x) d x.

Plot the function and its derivative on the interval [0,2].

(10 pts) Function f(x) is invertible, odd, monotonic, and its graph is below (the function is Tanh(x)):

If f has horizontal asymptotes of ±1, describe the domain and range of its inverse  (f^(-1)).

Carefully add the graph of its inverse,  (f^(-1)), to the plot above.

Given that f((3/2))=(9/10), and that (f^( '))((3/2))=(1/8); write the equations of the tangents line to f at  x=(3/2) and to (f^(-1)) at  x=(9/10), in point-slope form. [Note that these were approximations, amenable to working by hand -- use the correct values for Tanh and its derivative in this case].

Carefully draw both tangent lines on the axes above.

(10 pts) Compute the integrals.

∫x (e^(-x)) d x

∫(1/(Sqrt[4-(x^2)]))d x

(10 pts) Determine, by any of our methods, whether the following series converge or not, and, if they converge, whether they do so absolutely or conditionally:

(Underoverscript[Σ, (k=1), ∞])((sin(k))/(1+(k^2)))

(Underoverscript[Σ, (k=1), ∞])(((((-1))^k)(Sqrt[k]))/(k+1))

(10 pts) We learned that the Maclaurin series for cos(x) is(Underoverscript[Σ, (k=0), ∞])(((((-1))^k)(x^(2k)))/((2k)!)).

Determine its interval of convergence using a ratio test.

Use the Taylor remainder theorem to bound the error we make in estimating cos((π/4)) with the corresponding (8^th)degree Taylor polynomial. You do not actually need to compute the value of the bounding error -- but you must explain how you bound the appropriate derivative, and put everything together -- everything up to where you’d need a calculator.

Write the Maclaurin series for cos((Sqrt[x])).

(10 pts) Consider the definite integral
	(∫_0^5)ln(2+(e^(3x)))d x.

Graph the two derivatives for the integrand f(x)=ln(2+(e^(3x))) needed to estimate the error for the integral.

Determine a formula for n for which the following methods would estimate the integral with absolute error less than 0.001.

midpoint method

Simpson’s rule

Suppose we’d made a mistake, and the integral was actually (∫_0^5)ln(2*(e^(3x)))d x. Compute the integral analytically.

(10 pts) Determine, by any of our methods, the smallest value of n such that the error in computing an approximation to the infinite series S with the partial sum (Subscript[S, n])  will be within 0.0001 of the correct value.

(Subscript[S, n]) =  (Underoverscript[Σ, (k=1), n])k (e^(-k)).

(Subscript[S, n]) = (Underoverscript[Σ, (k=1), n])(1/(2^(k+3)))

(10 pts) Consider the parametric equations, given by x(t)=(1/2)(t^2) and y(t)=(1/3)(t^3).

Eliminate the parameter t to write a Cartesian equation for the curve.

Write and evaluate an integral for the length of the curve over the interval t∈[0,1].

Suppose the equations had been x(t)=3(e^t) and y(t)=6(e^t), instead. The integral becomes easy. Why? Describe the parametric motion on the  interval t∈[0,1].

(10 pts) Consider the vector  (Overscript[v, ->])=⟨4,-2,0⟩

Compute the exact length of  (Overscript[v, ->]) (that is, the norm of  (Overscript[v, ->]), or | (Overscript[v, ->])|).

Create a unit vector pointing in the direction of  (Overscript[v, ->]).

Find a vector (Overscript[u, ->]) perpendicular to  (Overscript[v, ->]) (and show that it is perpendicular by computing the dot product).

Use the cross product to find a vector perpendicular to both  (Overscript[u, ->]) and  (Overscript[v, ->]).

(10 pts) Consider the polar curve r=(Sqrt[cos(θ)+1]) . Graph it, to produce this plot:

What is the area enclosed by the curve on the interval -(π/2)≤θ≤π?

Sketch the area we computed above, and estimate it (roughly). Does your estimate agree with your calculation?