Question 1: Limits and Continuity

Question:
Evaluate the following limit, or show that it does not exist:

[ \lim_{{x \to 2}} \frac{{x^2 - 4}}{{x - 2}} ]

Solution:
To find the limit, we can simplify the expression:

[ \lim_{{x \to 2}} \frac{{x^2 - 4}}{{x - 2}} = \lim_{{x \to 2}} \frac{{(x - 2)(x + 2)}}{{x - 2}} ]

Canceling out the common factor, we get:

[ \lim_{{x \to 2}} (x + 2) = 2 + 2 = 4 ]

So, (\lim_{{x \to 2}} \frac{{x^2 - 4}}{{x - 2}} = 4).

Question 2: Derivatives and Optimization

Question:
A rectangular box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 10 cm by 15 cm by cutting equal squares from each corner and folding up the sides. Find the dimensions of the box that will maximize its volume.

Solution:
Let (x) be the side length of the squares cut from each corner. The dimensions of the box in terms of (x) are:

[ V(x) = x(10 - 2x)(15 - 2x) ]

To maximize (V(x)), we find its critical points by taking the derivative and setting it equal to zero:

[ \frac{{dV}}{{dx}} = 4x^3 - 50x^2 + 150x ]

Setting (\frac{{dV}}{{dx}} = 0), we find (x = 1.5) cm. Therefore, the dimensions of the box that maximize its volume are (7 \, \text{cm} \times 12 \, \text{cm} \times 12 \, \text{cm}).

Question 3: Integration and Area

Question:
Evaluate the definite integral:

[ \int_{{0}}^{{\pi}} x \sin(x) \, dx ]

Solution:
To evaluate the integral, use integration by parts:

[ \int u \, dv = uv - \int v \, du ]

Let (u = x) and (dv = \sin(x) \, dx). Then, find (du) and (v). Apply the integration by parts formula and evaluate the definite integral.

Question 4: Sequences and Series

Question:
Consider the series (\sum_{{n=1}}^{{\infty}} \frac{{3^n + 2}}{{n^2 + 1}}). Determine whether the series converges or diverges.

Solution:
Apply the ratio test to determine the convergence of the series:

[ \lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} ight| ]

where (a_n = \frac{{3^n + 2}}{{n^2 + 1}}). If the limit is less than 1, the series converges.

Question 5: Multivariable Calculus

Question:
Find the critical points of the function (f(x, y) = x^2 + y^2 - 2xy + 4x) and determine whether each critical point is a minimum, maximum, or saddle point.

Solution:
To find the critical points, set the partial derivatives of (f) with respect to (x) and (y) equal to zero and solve the system of equations. Then, use the second derivative test to classify each critical point as a minimum, maximum, or saddle point.