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10th Edition

Soo T. Tan

Publisher: Brooks Cole

ISBN: 9781285464640

Chapter 3.2, Problem 51E

Textbook Problem

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A straight line perpendicular to and passing through the point of tangency of the tangent line is called the *normal* to the curve at that point. Find the equation of the tangent line and the normal to the curve

*y* =

at the point(1,

To determine

**To find:** The equation of the tangent line and the normal to the curve

**Given information:**

The equation of curve is

**Formula used**:

The quotient rule of differentiation is,

The above formula is defined when the denominator

**Calculation:**

**Section1:**

The derivative of the given curve gives the slope of the tangent line so use the quotient rule of differentiation for the given function to get the derivative function

Further solve the above equation for

The derivative function gives the slope of line that is
*x* in the equation (1) to get the value of slope.

The formula of point-slope form of the line is,

The tangent line has slope
*m*, 1 for

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

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What is the second derivative of a function f?...Ch. 3.5 - If s = f(t) gives the position of an object moving...Ch. 3.5 - Suppose f(t) measures the population of a country...Ch. 3.5 - Suppose f(t) measures the population of a country...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 1-20, find the first and second...Ch. 3.5 - In Exercises 21-28, find the third derivative of...Ch. 3.5 - In Exercises 21-28, find the third derivative of...Ch. 3.5 - In Exercises 21-28, find the third derivative of...Ch. 3.5 - In Exercises 21-28, find the third derivative of...Ch. 3.5 - In Exercises 21-28, find the third derivative of...Ch. 3.5 - In Exercises 21-28, find the third derivative of...Ch. 3.5 - In Exercises 21-28, find the third derivative of...Ch. 3.5 - In Exercises 21-28, find the third derivative of...Ch. 3.5 - acceleration of a falling object During the...Ch. 3.5 - acceleration of a car The distance s (in feet)...Ch. 3.5 - auto transmissions .and fuel economy In trying to...Ch. 3.5 - Alzheimer'S Disease As baby boomers enter their...Ch. 3.5 - Crime Rates The number of major crimes committed...Ch. 3.5 - gdp of a developing country A developing countrys...Ch. 3.5 - sales of vinyl records Vinyl records (LPs) almost...Ch. 3.5 - Median Age of U.S. Population The median age (in...Ch. 3.5 - Obesity in America The body mass index (BMI)...Ch. 3.5 - Test Flight of a Vtol In a test flight of the...Ch. 3.5 - Air Purification During testing of a certain brand...Ch. 3.5 - WORKING MOTHERS The percentage of mothers who work...Ch. 3.5 - AGING POPULATION The population of Americans age...Ch. 3.5 - In Exercises 42-46, determine whether the...Ch. 3.5 - In Exercises 42-46, determine whether the...Ch. 3.5 - In Exercises 42-46, determine whether the...Ch. 3.5 - In Exercises 42-46, determine whether the...Ch. 3.5 - In Exercises 42-46, determine whether the...Ch. 3.5 - LetC(x) be the average cost function associated...Ch. 3.5 - Let f be the function defined by the rule f(x) =...Ch. 3.5 - Construct a function f that has derivatives of...Ch. 3.5 - Show that a polynomial function has derivatives of...Ch. 3.5 - In Exercises 1-8, find the value of the second...Ch. 3.5 - In Exercises 1-8, find the value of the second...Ch. 3.5 - In Exercises 1-8, find the value of the second...Ch. 3.5 - In Exercises 1-8, find the value of the second...Ch. 3.5 - In Exercises 1-8, find the value of the second...Ch. 3.5 - In Exercises 1-8, find the value of the second...Ch. 3.5 - In Exercises 1-8, find the value of the second...Ch. 3.5 - In Exercises 1-8, find the value of the second...Ch. 3.6 - a. Suppose the equation F(x, y) = 0 defines y as a...Ch. 3.6 - Suppose the equation xg(y) + yf(x) = 0, where f...Ch. 3.6 - In your own words, describe what a related-rates...Ch. 3.6 - Give the steps that you would use to solve a...Ch. 3.6 - In Exercises 1-8, find the derivative dy/dx (a) by...Ch. 3.6 - In Exercises 1-8, find the derivative dy/dx (a) by...Ch. 3.6 - In Exercises 1-8, find the derivative dy/dx (a) by...Ch. 3.6 - In Exercises 1-8, find the derivative dy/dx (a) by...Ch. 3.6 - In Exercises 1-8, find the derivative dy/dx (a) by...Ch. 3.6 - In Exercises 1-8, find the derivative dy/dx (a) by...Ch. 3.6 - In Exercises 1-8, find the derivative dy/dx (a) by...Ch. 3.6 - In Exercises 1-8, find the derivative dy/dx (a) by...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 9-30, find dy/dx by implicit...Ch. 3.6 - In Exercises 31-34, find an equation of the...Ch. 3.6 - In Exercises 31-34, find an equation of the...Ch. 3.6 - In Exercises 31-34, find an equation of the...Ch. 3.6 - In Exercises 31-34, find an equation of the...Ch. 3.6 - In Exercises 35-38, find the second derivative...Ch. 3.6 - In Exercises 35-38, find the second derivative...Ch. 3.6 - In Exercises 35-38, find the second derivative...Ch. 3.6 - In Exercises 35-38, find the second derivative...Ch. 3.6 - Volume of a Cylinder The volume of a...Ch. 3.6 - Distance Between Two Cars A car leaves an...Ch. 3.6 - Effect of Price on Demand for Tires Suppose the...Ch. 3.6 - Effect of Price on Supply of Tires Suppose the...Ch. 3.6 - Effect of Price on Demand for Headphones The...Ch. 3.6 - Effect of Price on Supply of Eggs Suppose the...Ch. 3.6 - Effect of Supply on price of Eggs Refer to...Ch. 3.6 - Elasticity of Demand for Ink-Jet Cartridges The...Ch. 3.6 - Elasticity of demand for CDs The demand function...Ch. 3.6 - Production Function for a Furniture Company The...Ch. 3.6 - Output of a Country Suppose that the output Q of a...Ch. 3.6 - Volume of a Cube The volume V of a cube with sides...Ch. 3.6 - Oil Spills In calm waters, the oil spilling from...Ch. 3.6 - Distance Between Two Ships Two ships leave the...Ch. 3.6 - Oil. Spills In calm waters, the oil spilling from...Ch. 3.6 - Distance Between Two Cars a car leaves an...Ch. 3.6 - Distance Between Two Cars a car leaves an...Ch. 3.6 - Watching a Helicopter Take off At a distance of 50...Ch. 3.6 - Watching a rowing Race a spectator watches a...Ch. 3.6 - Docking A Boat A boat is pulled toward a dock by...Ch. 3.6 - A Melting Snowball Assume that a snowball is in...Ch. 3.6 - Blowing soap Bubbles Carlos is blowing air into a...Ch. 3.6 - Coast Guard Patrol search mission The pilot of a...Ch. 3.6 - FILLING A COFFEE POT A coffee pot in the form of a...Ch. 3.6 - Movement of a Shadow A 6-ft tall man is walking...Ch. 3.6 - A Sliding Ladder A 20-ft ladder leaning against a...Ch. 3.6 - A Sliding Ladder The base of a 13-ft ladder...Ch. 3.6 - Flow of Water from a Tank Water flows from a tank...Ch. 3.6 - Volume of a Gas In an adiabatic process (one in...Ch. 3.6 - Mass of a Moving Particle The mass m of a panicle...Ch. 3.6 - In Exercises 69-74. determine whether the...Ch. 3.6 - In Exercises 69-74. determine whether the...Ch. 3.6 - In Exercises 69-74. determine whether the...Ch. 3.6 - In Exercises 69-74. determine whether the...Ch. 3.6 - In Exercises 69-74. determine whether the...Ch. 3.6 - In Exercises 69-74. determine whether the...Ch. 3.7 - 1. If y = y(x), what is the differential of x?...Ch. 3.7 - Refer to the following figure. a. Express A, B,...Ch. 3.7 - Suppose the population of a city at time t0 is...Ch. 3.7 - Suppose that the total profit of a company at time...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - In Exercises 1-14, find the differential of the...Ch. 3.7 - Let f be the function defined by y = f(x) = x2 1...Ch. 3.7 - Let f be the function defined by y = f(x) 5 3x2 ...Ch. 3.7 - Let f be the function defined by y=f(x)=1x a. Find...Ch. 3.7 - Let f be the function defined by y=f(x)=2x+1 a....Ch. 3.7 - In Exercises 19-26, use differentials to...Ch. 3.7 - In Exercises 19-26, use differentials to...Ch. 3.7 - In Exercises 19-26, use differentials to...Ch. 3.7 - In Exercises 19-26, use differentials to...Ch. 3.7 - In Exercises 19-26, use differentials to...Ch. 3.7 - In Exercises 19-26, use differentials to...Ch. 3.7 - In Exercises 19-26, use differentials to...Ch. 3.7 - In Exercises 19-26, use differentials to...Ch. 3.7 - Use a differential to approximate 4.02+14.02.Ch. 3.7 - Use a differential to approximate...Ch. 3.7 - Error Estimation The length of each edge of a cube...Ch. 3.7 - Estimating the Amount of Paint Required A coat of...Ch. 3.7 - Estimating the Amount of Rust-Proofer Required A...Ch. 3.7 - Growth of a CANCEROUS tumor The volume of a...Ch. 3.7 - Unclogging arteries Research done in the 1930s by...Ch. 3.7 - effect of capital expenditures on gdp An economist...Ch. 3.7 - learning curves The length of time (in seconds) a...Ch. 3.7 - effect of advertising on profits The relationship...Ch. 3.7 - effect of mortgage rates on housing starts A study...Ch. 3.7 - Effect of Supply of Radios on Price The supply...Ch. 3.7 - Effect of demand for smoke alarms on price The...Ch. 3.7 - Surface area of an animal Animal physiologists use...Ch. 3.7 - Forecasting Profits The management of Trappee and...Ch. 3.7 - Forecasting Commodity Prices A certain countrys...Ch. 3.7 - Demand for Tires The management of Titan Tire...Ch. 3.7 - Air Pollution The amount of nitrogen dioxide, a...Ch. 3.7 - Effect of Community Investment on Crime A...Ch. 3.7 - Financing a Home The Millers are planning to buy a...Ch. 3.7 - Investments Lup deposits a sum of 10,000 into an...Ch. 3.7 - Defined Benefit accounts Ian, who is...Ch. 3.7 - In Exercises 49 and 50, determine whether the...Ch. 3.7 - In Exercises 49 and 50, determine whether the...Ch. 3.7 - In Exercises 1-6, use dy to approximate y for the...Ch. 3.7 - In Exercises 1-6, use dy to approximate y for the...Ch. 3.7 - In Exercises 1-6, use dy to approximate y for the...Ch. 3.7 - In Exercises 1-6, use dy to approximate y for the...Ch. 3.7 - In Exercises 1-6, use dy to approximate y for the...Ch. 3.7 - In Exercises 1-6, use dy to approximate y for the...Ch. 3.7 - Calculating Mortgage Payments Refer to Example 2....Ch. 3.7 - Estimating THE Area ok a Ring of Neptune The ring...Ch. 3 - a. If c is a constant, then ddx(c)=__________. b....Ch. 3 - a. The Product Rule states that...Ch. 3 - a. The Chain Rule states that if h(x) = g[f(x)],...Ch. 3 - If C, R, P, and C denote the total cost function,...Ch. 3 - a. If f is a differentiable demand function...Ch. 3 - Suppose a function y = f(x) is defined implicitly...Ch. 3 - In a related-rates problem, we are given a...Ch. 3 - Let y = f(t) and x = g(t). If x2 + y2 = 4, then...Ch. 3 - a. If a variable quantity x changes from x1 to x2,...Ch. 3 - If y = f(x), where f is a differentiable function,...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-20, find the first and second...Ch. 3 - In Exercises 1-20, find the first and second...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 1-30, find the derivative of the...Ch. 3 - In Exercises 31-36, find the second derivative of...Ch. 3 - In Exercises 31-36, find the second derivative of...Ch. 3 - In Exercises 31-36, find the second derivative of...Ch. 3 - In Exercises 31-36, find the second derivative of...Ch. 3 - In Exercises 31-36, find the second derivative of...Ch. 3 - In Exercises 31-36, find the second derivative of...Ch. 3 - In Exercises 37-42, find dy/dx by implicit...Ch. 3 - In Exercises 37-42, find dy/dx by implicit...Ch. 3 - In Exercises 37-42, find dy/dx by implicit...Ch. 3 - In Exercises 37-42, find dy/dx by implicit...Ch. 3 - In Exercises 37-42, find dy/dx by implicit...Ch. 3 - In Exercises 37-42, find dy/dx by implicit...Ch. 3 - Find the differential of f(x)=x2+1x2Ch. 3 - Find the differential of f(x)=1x3+1Ch. 3 - Let f be the function defined by f(x)=2x2+4 a....Ch. 3 - Use a differential to approximate 26.83Ch. 3 - Let f(x) = 2x3 3x2 16x + 3. a. Find the points...Ch. 3 - Find an equation of the tangent line to the graph...Ch. 3 - Find an equation of the tangent line to the graph...Ch. 3 - Find the third derivative of the function...Ch. 3 - SALES OF DSPs The annual sales of digital signal...Ch. 3 - U.k. Digital Video Viewers Digital video viewing...Ch. 3 - adult obesity In the United States, the percentage...Ch. 3 - Sales of Cameras The shipments of Lica digital...Ch. 3 - Gdp of a Country The gross domestic product (GDP)...Ch. 3 - Population Growth in A Suburb The population of a...Ch. 3 - Best-Selling Novel The number of copies of a...Ch. 3 - Cable TV Subscribers The number of subscribers to...Ch. 3 - Cost of Wireless Phone Calls As cellular phone...Ch. 3 - Male Life Expectancy Suppose the life expectancy...Ch. 3 - Cost of Producing DVDs The total weekly cost in...Ch. 3 - Supply Function for Satellite Radios The supply...Ch. 3 - Demand Function for Electric Shavers The demand...Ch. 3 - Marginal Cost The total daily cost (in dollars)...Ch. 3 - Demand for Smartphones The marketing department of...Ch. 3 - Demand for Photocopying Machines The weekly demand...Ch. 3 - Marginal Average Cost The Custom Office makes a...Ch. 3 - Elasticity of Demand The demand equation for a...Ch. 3 - Elasticity of Demand The demand equation for a...Ch. 3 - Elasticity OF Demand The demand equation for a...Ch. 3 - Elasticity of Demand The demand equation for a...Ch. 3 - GDP of a Country The GDP of a country from the...Ch. 3 - Motion of an Object The position of an object...Ch. 3 - Find the derivative of f(x) = 2x3 3x1/3 + 5x2/3.Ch. 3 - Differentiateg(x)=x2x21Ch. 3 - Find dydx if y=2x+1x2+x+1Ch. 3 - Find the first three derivatives of, f(x)=1x+1Ch. 3 - Find dydx given that xy2 x2y + x3 = 4.Ch. 3 - Let y=xx2+5 a. Find the differential of y. b. If x...

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