Chapter 4 Review Test
Hey everyone! Here's the review test for Chapter 4. We'll be going over this in class on Monday, so make sure you give it a good try beforehand. This test covers important concepts we've learned, including finding minimum and maximum values, analyzing functions, and solving optimization problems. Good luck, and remember to show your work!
- Find the minimum/maximum value for $y = 2x^4 - 8x$.
Hint: Remember to find the critical points by taking the derivative and setting it equal to zero. Then, use the first or second derivative test to determine if they are minima or maxima. - Find the domain restrictions and any asymptotes for $f(x) = \sqrt{x^2 + 2x} - x$.
Hint: Consider where the expression under the square root is non-negative for domain restrictions. For asymptotes, analyze the behavior of the function as $x$ approaches infinity. - Find the domain and over which interval the function is concave up and concave down for $y = x(\ln(x))^2$.
Hint: You'll need to find the second derivative, $y''$, to determine concavity. Set $y'' > 0$ for concave up and $y'' < 0$ for concave down. Don't forget to consider the domain of the natural logarithm. - Find the intervals that $f(x) = x^2(x - 6)$ is increasing or decreasing.
Hint: Find the first derivative, $f'(x)$, and determine where $f'(x) > 0$ (increasing) and $f'(x) < 0$ (decreasing). - Find the point of inflection for $f(x) = x^3 - 3x^2 - 6x - 3$.
Hint: A point of inflection occurs where the second derivative changes sign. Find $f''(x)$ and determine where it equals zero or is undefined. - Solve $\lim_{x \to \frac{\pi}{4}} \frac{\cos(x) - \sin(x)}{\tan(x) - 1}$.
Hint: Try to manipulate the expression to avoid an indeterminate form (e.g., using L'Hôpital's Rule or trigonometric identities). - Solve $\lim_{x \to 1} \frac{\sin(x-1)}{x^3 + 2x - 3}$.
Hint: Notice that as $x$ approaches 1, both the numerator and denominator approach 0. Consider L'Hopital's rule. - A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $10 per linear foot to install and the farmer is not willing to spend more than $7000, find the dimensions for the plot that would enclose the most area.
Hint: This is an optimization problem. Let the length of the side parallel to the barn be $y$ and the length of the other two sides be $x$. You will have a cost constraint and need to maximize the area. - A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle, labeled $x$.) A window with the shape of a rectangle surmounted by a semicircle. The diameter of the semicircle is equal to the width of the rectangle. The rectangle has width $x$. If the perimeter of the window is 8 feet, find the exact value of $x$ (in ft) so that the greatest possible amount of light is admitted.
Hint: Express the perimeter of the window in terms of $x$ and the height $h$ of the rectangle. Solve for $h$ in terms of $x$, then express the area (rectangle + semicircle) in terms of $x$ only. Maximize the area. - A fence 8 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length (in feet) of the shortest ladder that will reach from the ground over the fence to the wall of the building? (Round your answer to two decimal places.)
Hint: Use similar triangles to set up a relationship between the ladder's length and its distance from the building. Minimize the ladder length using calculus.
Remember to review your notes and practice similar problems. See you in class on Monday!