The revenue is measured in dollars. Find the marginal profit for producing x units. The profit is measured in dollars. The cost C in dollars of producing x units of a product is given by C 3. Find the additional cost when the production increases from 9 t o The profit in dollars from selling x units of calculus textbooks is given by p 0. Find the additional profit when the sales increase from to units.
Round your answer to two decimal places. The profit in dollars from selling x units of calculus textbooks is given by x Round your answer p 0. Find the marginal profit when to two decimal places.
The population P in thousands of Japan from through can be modeled by P Determine the population growth rate, dP dt. A dP dt Find the profit P as a function of x, the number of glasses of lemonade sold. Find the marginal profit when glasses of lemonade are sold and when glasses of lemonade are sold. Use the product Rule to find the derivative of the function f x x x 2 3.
A 1, Product Rule. Ans: A Find the point s , if any, at which the graph of f has a horizontal tangent line. A population of bacteria is introduced into a culture. Find an equation of the tangent line to the graph of f at the given point. Find the rate of change 45 t 2 of the population when t 4. Find the derivative of the given function. Simplify and express the answer using positive exponents only. Ans: D Ans: E At the end of 4 years, the balance is A 1.
A The value V of a machine t years after it is purchased is inversely proportional to the square root of t 5. Find the rate of depreciation when t 2. A — Find the third derivative of the function. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The product rule is Ans: B. Find the third derivative. Find the value g 4 for the function g t 3t 8 6t 6 1. Find the indicated derivative.
A brick becomes dislodged from the Empire State Building at a height of feet and falls to the sidewalk below. Write the position s t , velocity v t , and acceleration a t as functions of time. Find y implicitly for 6 x9 y 9 3. Find the rate of change of x with respect to p. Find dy dx implicitly and explicitly the explicit functions are shown on the graph and show that the results are equivalent.
Use the graph to estimate the slope of the tangent line at the labeled point. Most theorems are proved, left as an exercise, or left unproved when the proof is too difficult.
When a proof is difficult, we try to give an intuitive explanation to make the result plausible before going on to the next topic. In some cases, we give a sketch of a proof, in which case we explain why it is a sketch and not a rigorous proof. The focus is still on understanding the concepts of calculus. While some see the emphasis on clear, rigorous presentation as being a distraction to understanding calculus, we see the two as complementary. Students are more likely to grasp the concepts of calculus if terms are clearly defined and theorems are clearly stated and proved.
A Brief Text The ninth edition continues to be the briefest of all the successful mainstream calculus texts. We have tried to prevent the text from ballooning up ward with new topics and alternative approaches. In less than BOO pages, we cover the major topics of calculus, including a preliminary chapter, and the material from limits to vector calculus.
In the last few decades, students have developed some bad habits. They prefer not to read the textbook. They want to find the appropriate worked-out example so it can be matched to their homework problem. Our goal with this text continues to be to keep calculus as a course focused on some few basic ideas centered around words, formulas, and graphs.
Solving problem sets, while crucial to developing mathematical and problem-solving skills, should not overshadow the goal of understanding calculus. Concepts Review Problems To encourage students to read the textbook with understanding, we begin every problem set with four fill-in-the-blank items These test the mastery of the basic vocabulary, understanding of theorems, and ability to apply the concepts in the simplest settings.
Students should respond to these items before proceeding to the later problems. We encourage this by giving immediate feedback; the correct answers are given at the end of the problem set.
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