Calculus 1 Midterm Part 2 | Implicit Differentiation & Related Rates

Calculus 1 Midterm Part 2 | Implicit Differentiation & Related Rates Alternate Titles (55–70 chars): 1. Calculus 1 Midterm Solved Part 2 | Related Rates & Growth Models (63) 2. Calc 1 Midterm Part 2 | Implicit Differentiation, Related Rates (62) 3. Calculus 1 Midterm Walkthrough Part 2 | Full Solutions (54) 4. Calculus 1 Midterm Part 2 | Related Rates, e^kt & Marginal Cost (63) Description: I solve the second half of my real Calculus 1 midterm, problem by problem, with full step-by-step solutions. This part covers implicit differentiation (including y″), related rates with a leaking conical tank, exponential growth models, and marginal cost word problems. Watch Part 1 first if you haven't: [   • Calculus 1 Midterm Review | Problems and S...  ] Topics covered: Implicit differentiation and finding the second derivative y″ Converting the quotient rule into a product rule to avoid ugly fractions Related rates — leaking conical tank (V = ⅓πr²h, substituting r = h/2) Why the rate comes out negative when a quantity is decreasing Chain rule + product rule with cos(x² + y²) Isolating y′ in implicit equations Exponential growth models: U(t) = U₀e^(kt), solving for k with ln Instantaneous rate of change vs. average rate of change Average cost and marginal cost functions Using marginal cost to decide whether to produce one more unit Timestamps: 00:00 Intro & recap of Part 1 00:17 Implicit differentiation — solving for y″ 02:00 Converting quotient rule to product rule 04:38 Simplifying the second derivative 05:56 Related rates — leaking conical tank 07:29 Setting up V = ⅓πr²h and substituting r = h/2 11:17 Solving for h′ at 7 m (and why it's negative) 13:04 Implicit differentiation with cos(x² + y²) 14:27 Product rule on the right side 16:18 Isolating y′ 17:00 Word problem — exponential user growth model 17:23 Part A: building U(t) = U₀e^(kt) 20:06 Part B: predicting users in 2019 20:34 Part C: instantaneous growth rate in 2016 22:28 Part D: average rate of change 2016–2018 24:46 Word problem — food truck cost function 25:06 Part A: average cost of first 200 burgers 26:41 Part B: marginal cost function 27:22 Part C: approximate cost of the 201st burger 28:41 Part D: profit per burger at $6 29:59 Part E: should they make one more? 31:17 Final tips & good luck Course: Calculus 1 (Calc I / MATH 1010) Level: First-year university & AP Calculus AB One tip from the video: memorize the cone volume formula — a lot of professors won't give it to you on the exam. Try each problem yourself before watching the solution, then check your method against mine. Let me know in the comments what you'd like me to solve next. #calculus1 #relatedrates #implicitdifferentiation