Calculus For Biology And Medicine 3rd Edition Answers
Calculus is an integral part of biology and medicine, and it is essential to have a good understanding of the subject to excel in these fields. The third edition of Calculus for Biology and Medicine by Claudia Neuhauser is a comprehensive guide that helps students grasp the concepts of calculus and apply them to various problems in biology and medicine. However, many students struggle with the exercises and problems in the book and need answers to check their work. In this article, we will provide answers to some of the questions in Calculus for Biology and Medicine 3rd Edition.
Chapter 1: Functions and Models
In this chapter, students are introduced to the concept of functions and models. They learn about linear functions, exponential functions, logarithmic functions, and the laws of logarithms. The exercises in this chapter test their understanding of these concepts and their ability to apply them to real-world problems.
Q1. Find the slope of the line that passes through the points (2, 3) and (4, 5).
A1. The slope of the line is (5-3)/(4-2) = 1.
Q2. Solve for x: log4(x+2) = 2.
A2. x+2 = 42 = 16. Therefore, x=14.
Chapter 2: Limits and Derivatives
This chapter introduces the concept of limits and derivatives. Students learn about the limit laws, continuity, and the definition of the derivative. They also learn to differentiate various functions like polynomials, exponential functions, and logarithmic functions.
Q1. Find the derivative of f(x) = x3 - 2x2 + 5x - 1.
A1. The derivative of f(x) is f'(x) = 3x2 - 4x + 5.
Q2. Find the equation of the tangent line to the graph of f(x) = 2x2 - 3x + 1 at the point (1, 0).
A2. The slope of the tangent line is f'(1) = 4-3 = 1. Therefore, the equation of the tangent line is y = 1(x-1) + 0 = x-1.
Chapter 3: Additional Applications of the Derivative
In this chapter, students learn about additional applications of the derivative. They learn to use the derivative to find the maximum and minimum values of a function, and to solve optimization problems.
Q1. Find the maximum value of the function f(x) = x3 - 3x2 + 4x + 1 on the interval [0, 3].
A1. The critical points of f(x) are x=1 and x=2. We evaluate f(x) at these points and at the endpoints of the interval: f(0)=1, f(1) = 3, f(2) = 5, and f(3) = 19. Therefore, the maximum value of f(x) on the interval is 19.
Q2. A rectangular box with a square base and no top is to be made from a square piece of cardboard of side 30 cm by cutting out a square from each corner and folding up the sides. What is the maximum volume of such a box?
A2. Let x be the length of the side of the square that is cut out from each corner. Then the height of the box is 30-2x, and the volume is V(x) = x(30-2x)2. We find the critical point of V(x) by differentiating and setting the derivative equal to zero: V'(x) = -12x2 + 180x - 1800 = 0. Solving this equation, we get x=5 or x=15. We evaluate V(x) at these points and at the endpoints of the interval [0, 15]: V(0)=0, V(5)=1250, V(15)=0, and V(30)=0. Therefore, the maximum volume of the box is 1250 cm3.
Chapter 4: Exponential and Logarithmic Functions
In this chapter, students learn about exponential and logarithmic functions. They learn to differentiate these functions and to solve problems involving growth and decay.
Q1. Find the derivative of f(x) = e2x - ln(x).
A1. The derivative of f(x) is f'(x) = 2e2x - 1/x.
Q2. A radioactive substance has a half-life of 10 days. If there are initially 100 grams of the substance, how much will be left after 30 days?
A2. After 10 days, half of the substance will be left, so there will be 50 grams. After 20 days, another half will be left, so there will be 25 grams. After 30 days, another half will be left, so there will be 12.5 grams.
Chapter 5: Integration and Its Applications
This chapter introduces the concept of integration and its applications. Students learn to find the antiderivative of a function, to evaluate definite integrals, and to solve problems involving areas and volumes.
Q1. Evaluate the definite integral ∫02 (x2 + x + 1) dx.
A1. We evaluate the integral by finding the antiderivative of the integrand and evaluating it at the limits of integration: ∫02 (x2 + x + 1) dx = [(1/3)x3 + (1/2)x2 + x] 02 = (8/3) + 2 + 2 = 14/3.
Q2. Find the volume of the solid obtained by rotating the region bounded by y=x2, y=0, and x=2 about the x-axis.
A2. We use the disk method to find the volume: V = ∫02 π(x2)2 dx = π∫02 x4 dx = π(32/5) = 6.4π.
Conclusion
Calculus for Biology and Medicine 3rd Edition is an excellent resource for students who want to learn calculus and its applications in biology and medicine. The exercises and problems in the book are challenging, and the answers provided in this article will help students check their work and gain a better understanding of the subject. By mastering the concepts in this book, students will be well-equipped to tackle the calculus problems that arise in their future studies and careers.