Calculus I (MAT 145) Dr. Day Wednesday April 10, 2019 Chapter 4: Using All Your Derivative Knowledge! Absolute and Relative Extremes What is a “critical number?” Increasing and Decreasing Behavior of Functions Connecting f and f’ Concavity of Functions: A function’s curvature Connecting f and f” Graphing a Function: Putting it All together! Max-Mins Problems: Determine Solutions for Contextual Situations Finally . . . What if We Reverse the Derivative Process? Monday, April 8, 2019 MAT 145

Steps for Optimizing Understand the problem: What is given? What is requested? Are there constraints? Diagram and variables: If appropriate for the context draw a diagram and label with variables. Define variables (by saying what the letters you are using represent, including units) and determine a reasonable span of values for those variables in the context of the problem. This will help you identify the domain of the independent variable. Determine quantity to be optimized and write an equation: Read the problem to determine what you have been asked to optimize. Express the quantity to be optimized (one of your variables), as a function of the other variable(s) in the problem. Identify constraint equation, if appropriate: If there are several variables, one may be constrained to be a function of the others. Write an equation that relates those variables. Combine with other equation previously defined, if appropriate. Use the closed interval method to identify the absolute maximum or minimum Answer the question! Monday, April 8, 2019 MAT 145

Bendable Wire A bendable wire measures 1000 cm in length. Write a function, call it A(x), to represent the sum of the areas of a circle and a square that result when that wire is cut at one point and the resulting two pieces of wire are used to create those shapes. Determine how to cut the wire so that the sum of the areas of the circle and the wire are a maximum. Monday, April 8, 2019 MAT 145

Bendable Wire Monday, April 8, 2019 MAT 145

Bendable Wire Note: The A’ function is a polynomial; so A’ is defined everywhere on its domain. Thus, the only critical numbers occur when A’ = 0. A(4000/(π+4)) ≈35,006, A(0)≈79,577, A(1000)=62,500. Thus, max area occurs when x=0 and entire wire is used to make a circle. Monday, April 8, 2019 MAT 145

Solve the follow optimization problem Solve the follow optimization problem. Show complete evidence and calculus justification. Include a drawing or a graph to represent the situation. Include evidence that shows you have considered any domain restrictions.   ____ 1 pt: labeled sketch, drawing, graph; ____ 1 pt: variables identified/described; ____ 1 pt: domain of independent variable; ____ 1 pt: statement of optimizing function; ____ 1 pt: constraint; ____ 1 pt: calculus evidence; ____ 1 pt: justify optimum; ____ 1 pt: consider all possibilities for critical points; ____ 1 pt: correct solution, units labeled Monday, April 8, 2019 MAT 145

Wednesday, Oct 31, 2018 MAT 145 Recovering Functions

Recovering Functions: Solutions p1 Solutions p2 Wednesday, Oct 31, 2018 MAT 145

Position, Velocity, Acceleration An object is moving in a positive direction when …. An object is moving in a negative direction when …. An object speeds up when …. An object slows down when …. An object changes directions when …. The average velocity over a time interval is found by …. The instantaneous velocity at a specific point in time is found by …. The net change in position over a time interval is found by …. The total distance traveled over a time interval is found by …. Wednesday, Oct 31, 2018 MAT 145

Position, Velocity, Acceleration An object is moving in a positive direction when v(t) > 0. An object is moving in a negative direction when v(t) < 0. An object speeds up when v(t) and a(t) share same sign. An object slows down when v(t) and a(t) have opposite signs. An object changes directions when v(t) = 0 and v(t) changes sign. The average velocity over a time interval is found by comparing net change in position to length of time interval (SLOPE!). The instantaneous velocity at a specific point in time is found by calculating v(t) for the specified point in time. The net change in position over a time interval is found by calculating the difference in the positions at the start and end of the interval. The total distance traveled over a time interval is found by first determining the times when the object changes direction, then calculating the displacement for each time interval when no direction change occurs, and then summing these displacements. Wednesday, Oct 31, 2018 MAT 145

Position, velocity, acceleration Velocity: rate of change of position Acceleration: rate of change of velocity Velocity and acceleration are signed numbers. Sign of velocity (pos./neg.) indicates direction of motion (right/left or up/down) When velocity and acceleration have the same sign (both pos. or both neg.), then object is speeding up. This is because object is accelerating the same direction that the object is moving. When velocity and acceleration have opposite sign (one positive and one negative), then object is slowing down. This is because object is accelerating the opposite direction that the object is moving. Wednesday, Oct 31, 2018 MAT 145

Derivative Patterns You Must Know The Derivative of a Constant Function The Derivative of a Power Function The Derivative of a Function Multiplied by a Constant The Derivative of a Sum or Difference of Functions The Derivative of a Polynomial Function The Derivative of an Exponential Function The Derivative of a Logarithmic Function The Derivative of a Product of Functions The Derivative of a Quotient of Functions The Derivatives of Trig Functions Derivatives of Composite Functions (Chain Rule) Implicit Differentiation Wednesday, Oct 31, 2018 MAT 145

Antiderivatives, Integrals, and Initial-Value Problems Knowing f ’, can we determine f ? General and specific solutions: The antiderivative. The integral symbol: Representing antiderivatives Initial-Value Problems: Transforming a general antiderivative into a specific function that satisfies the given information. Read this: “The antiderivative of 2x with respect to the variable x”

If we know a rate function . . . A particle moves along the x-axis. It’s velocity is given by v(t) = 2t2-3t+1 If we know that the particle is at location s = 3 at time t = 0, that is, that s(0) = 3, determine the position function s(t). What is the particle’s location at time t = 10? Wednesday, Oct 31, 2018 MAT 145

If we know a rate function . . . A car travels down the interstate at 60 mph. At milepost 240, the driver applies the brakes fully, resulting in a constant deceleration of 40 ft/sec2. What is the distance required for the car to come to a complete stop? Wednesday, Oct 31, 2018 MAT 145

If we know a rate function . . . Snow begins falling at midnight at a rate of 1 inch of snow per hour. It stops snowing at 6 am, 6 hours later. Write an accumulation function S(t), to describe the total amount of snow that had fallen by time t, where 0 ≤ t ≤ 6 hrs. Wednesday, Oct 31, 2018 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, Oct 31, 2018 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, Oct 31, 2018 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, Oct 31, 2018 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, Oct 31, 2018 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, Oct 31, 2018 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, Oct 31, 2018 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, Oct 31, 2018 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, Oct 31, 2018 MAT 145