What is the average value of 2t3

Limits and Continuity 3. Partial Differentiation 4. The Chain Rule 5. Directional Derivatives 6. Higher order derivatives 7. Maxima and minima 8. Lagrange Multipliers 15 Multiple Integration 1. Volume and Average Height 2. Double Integrals in Cylindrical Coordinates 3. Moment and Center of Mass 4. Surface Area 5. Triple Integrals 6. Cylindrical and Spherical Coordinates 7. Change of Variables 16 Vector Calculus 1. Vector Fields 2. Line Integrals 3. The Fundamental Theorem of Line Integrals 4.

Green's Theorem 5. Divergence and Curl 6. Vector Functions for Surfaces 7. Surface Integrals 8. Stokes's Theorem 9. The Divergence Theorem 17 Differential Equations 1. First Order Differential Equations 2. First Order Homogeneous Linear Equations 3. First Order Linear Equations 4. Approximation 5. Second Order Homogeneous Equations 6. Second Order Linear Equations 7. Basics 2. Differentiation 3. Figure Solution The velocity is the derivative of the position function:. Hint Find and look at the sign.

Population Change In addition to analyzing velocity, speed, acceleration, and position, we can use derivatives to analyze various types of populations, including those as diverse as bacteria colonies and cities. Estimating a Population The population of a city is tripling every 5 years. Solution Let be the population in thousands years from now.

Hint Use. Changes in Cost and Revenue In addition to analyzing motion along a line and population growth, derivatives are useful in analyzing changes in cost, revenue, and profit.

Applying Marginal Revenue Assume that the number of barbeque dinners that can be sold, , can be related to the price charged, , by the equation. In this case, the revenue in dollars obtained by selling barbeque dinners is given by for. Solution First, find the marginal revenue function:.

The actual revenue obtained from the sale of the st dinner is , or. Hint Use to approximate. Key Concepts Using , it is possible to estimate given and. The rate of change of position is velocity, and the rate of change of velocity is acceleration. Speed is the absolute value, or magnitude, of velocity.

The population growth rate and the present population can be used to predict the size of a future population. Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item. Find the velocity and acceleration functions. Determine the time intervals when the object is slowing down or speeding up.

Solution a. Speeds up: ; Slows down:. Find the velocity of the rocket 3 seconds after being fired. Find the acceleration of the rocket 3 seconds after being fired. Determine how long it takes for the ball to hit the ground. Determine the velocity of the ball when it hits the ground. Determine the velocity of the car when. Determine the velocity of the bird at sec. Determine the acceleration of the bird at sec. Determine the acceleration of the bird when the velocity equals 0.

Find the velocity of the potato after 0. Find the speed of the potato at 0. Determine when the potato reaches its maximum height. Find the acceleration of the potato at 0. Determine how long the potato is in the air. Determine the velocity of the potato upon hitting the ground. Determine the direction the train is traveling when. Determine the time intervals when the train is slowing down or speeding up.

Use the graph of the position function to determine the time intervals when the velocity is positive, negative, or zero. Sketch the graph of the velocity function. Use the graph of the velocity function to determine the time intervals when the acceleration is positive, negative, or zero. Determine the time intervals when the object is speeding up or slowing down. Find the marginal cost function. Find the marginal cost of manufacturing 12 food processors.

Find the actual cost of manufacturing the thirteenth food processor. Find the revenue function. Find the marginal revenue function. Find the marginal revenue at and. Find the exact profit from the sale of the thirtieth skateboard. Find the marginal profit function and use it to estimate the profit from the sale of the thirtieth skateboard. Find the revenue and marginal revenue functions.

Find and. Interpret the results. Find the profit and marginal profit functions. Find the rate of change function of the population function.

Find , and. Interpret what the results mean for the town. Find the rate of change of the number of bacteria. Interpret the results in b. Interpret what the answers imply about the bacteria population growth. Computer Science. Electrical Engineering. Mechanical Engineering. Advanced Math. Advanced Physics. Anatomy and Physiology. Earth Science. Social Science. Political Science. Literature Guides. Popular Textbooks. Want to see the step-by-step answer?

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Using average value to find average with respect to t and s. Ask Question. Asked 5 years, 1 month ago. Active 5 years, 1 month ago. Viewed 1k times. Zack Sloan Zack Sloan 13 2 2 bronze badges. Add a comment. Active Oldest Votes.