This is my study guide for a calc test. Can you answer all these questions with shown work and explanation?

Unit 1: Limits 8. Continuity 1. Evaluate the following limits. Remember to consider the limits from both sides. lin:(3.1rl - 2.1" +4) a. lim 1+ C. -2x3 +5 lim 3.x2 d. x- + 3x - 10 lim e x+5 Hint: Both factoring and L'Hopital's Rule are OK, though you must show why you need to use it. sin c lim I + 00 Hint: No proof required. If you don't remember the answer, use L'Hopital's Rule, though you must show why you need to use it.2. Consider [1(1) 2x231? and its graph. a. Find the equation for the instantaneous slope using the Product Rule. b. Use the derivative to nd the equation of the tangent line at the point (1, e). 0. Given that f'(x) = a} , find the differential dy in terms of dx. Consider f(x) = I3 - 2x2 and its graph. a. Use f'(x) and the 1st Derivative Test to find the local minimum and maximum of f(x). Use the graph above to help conrm your answer. b. Identify all regions in which f(x) is increasing and those in which it is decreasing. Use the graph above to help confirm your answer. 0. Use f"(x) to nd the inection point on the graph. Use the graph above to help confirm your answer. Unit 4: Applications of Derivatives 4. Consider a rectangle in which the length is always quadruple the size of the width. a. Find the rate at which the length increases if the width increases by 3 m/s. b. Find a general equation describing the rate at which the area expands with respect to its width. 0. Use your equation to find the rate at which the area expands if the width increases by 3 m/s when the width is 20 rn wide. 5. You are responsible for creating a fenced rectangular area for a mix of animals at a farm. The farmer demands that the rectangular area be divided into four quadrants to separate the chickens, sheep, pigs, and cattle, and even worse, he only gives you 1200 ft of fence to work with. Fortunately, the rectangular area can be braced against the ludicrously large barn (consider it infinitely long), which will save you the effort of building one side of the fenced rectangular area. a. Draw and label a picture of the scenario. b. Find the length and width that maximize the fenced rectangular area. 0. What is the actual area of the fenced region