I want a slightly more advanced calculus vs algebra test that allows me to answer questions throughout the video

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Welcome to this interactive calculus versus algebra test. Throughout this video, you'll be presented with questions that can be solved using either calculus or algebraic approaches. Question 1: Find the limit as x approaches 0 of sine of 3x divided by x. Your options are: A) 0, B) 1, C) 3, or D) Does not exist. Pause the video now to solve this problem, then continue to see the solution. Let's solve this problem using calculus. We can rewrite the limit as 3 times the limit of sine of 3x divided by 3x. Using the substitution u equals 3x, this becomes 3 times the limit of sine u over u as u approaches zero. We know from the fundamental limit that sine of u over u approaches 1 as u approaches zero. Therefore, our answer is 3 times 1, which equals 3. The correct answer is C. Notice on the graph how the function approaches the horizontal line y equals 3 as x approaches zero from both sides. Now let's move on to Question 2. Find the equation of the tangent line to the function f of x equals x squared minus 4x plus 5 at the point where x equals 2. Your options are: A) y equals negative 2x plus 5, B) y equals zero x plus 1, C) y equals 4x minus 7, or D) y equals negative 4x plus 9. This problem can be solved using either calculus by finding the derivative, or using algebra by finding the slope between two very close points. Pause the video now to solve this problem, then continue to see the solution. Let's solve this problem using calculus. First, we find the derivative of the function. The derivative of x squared minus 4x plus 5 is 2x minus 4. Evaluating the derivative at x equals 2 gives us 2 times 2 minus 4, which equals 0. This means the slope of the tangent line at x equals 2 is zero, so the tangent line is horizontal. To find the equation of this line, we need the y-coordinate of the point. Evaluating the original function at x equals 2 gives us 2 squared minus 4 times 2 plus 5, which equals 1. Therefore, the tangent line passes through the point (2,1) with a slope of 0. The equation of the tangent line is y equals 1, which matches option B: y equals zero x plus 1. Let's summarize what we've learned from these problems. Calculus provides powerful tools for solving limit and derivative problems that would be difficult using only algebraic methods. For limits, techniques like substitution and knowing fundamental limits are essential. When finding tangent lines, the derivative gives us the exact slope at a specific point. While algebraic approaches can sometimes provide alternative solutions, calculus often offers more direct and elegant methods. By practicing both calculus and algebraic approaches, you'll develop stronger mathematical problem-solving skills and gain a deeper understanding of the relationships between different areas of mathematics.