Calculus 1 Exam 3 Review Problems and Solutions (a lot about Optimization and Integrals)

1) True/False questions related to the Mean Value Theorem & optimization, 2) Fundamental Theorem of Calculus, 3) Linearity of integration, 4) L’Hopital’s Rule, 5) First Derivative Test and Second Derivative Test, 6) Family of functions optimization, 7) Integrate the velocity to get the change in position, 8) Related rates problem (sliding ladder), 9) Minimize surface area for fixed volume, 10) Differential equation, 11) Density function and cumulative distribution function, 12) Parametric curve speed and distance traveled (arc length), 13) Integrate the flow rate to find the change in volume, 14) Prove a function is constant with the Constant Function Theorem.

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(0:00) Video summary
(0:13) Mean Value Theorem and Corollaries (e.g., Increasing Function Theorem)
(4:07) Differentiate an integral (Fundamental Theorem of Calculus…Antiderivative Construction Theorem)
(6:11) Evaluate an integral (Fundamental Theorem of Calculus)
(7:52) Linearity property of definite integrals
(9:28) L’Hopital’s Rule
(12:54) First Derivative Test and Second Derivative Test (Optimization Problem: critical points and local maximum and minimum values)
(21:05) Two parameter family of functions optimization problem
(26:45) Find change in position from velocity (displacement is the integral of the velocity)
(31:14) Sliding ladder related rates problem
(36:52) Minimize the surface area of a closed cylinder when the volume is fixed (minimize the objective function subject to a constraint)
(41:49) Differential equations (verify that a function is a solution)
(43:47) Density function and cumulative distribution function
(49:00) Parametric equations: graph the parametric curve, find the speed, estimate the distance traveled
(59:25) Find volume by integrating flow rate (graphical problem)
(1:03:08) Prove a function is constant using the Constant Function Theorem (and the Chain Rule)

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