Math

CalculusCalculus of a Single VariableLinear and Quadratic Approximations The linear and quadratic approximations of a function f at x = a are P 1 ( x ) = f ' ( a ) ( x − a ) + f ( a ) and P 2 ( x ) = 1 2 f ″ ( a ) ( x − a ) 2 + f ' ( a ) ( x − a ) + f ( a ) In Exercises 119-120, (a) find the specified linear and quadratic approximations of f , (b) use a graphing utility to graph f and the approximations, (c) determine whether P 1 or P 2 b the better approximation, and (d) state how the accuracy changes as you move farther from x = a . f ( x ) = tan x ; a = π 4BuyFind*launch*

11th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337275361

Chapter 2.4, Problem 119E

Textbook Problem

Linear and Quadratic Approximations The linear and quadratic approximations of a function *f* at

**are**

**and**

**In Exercises 119-120, (a) find the specified linear and quadratic approximations of f, (b) use a graphing utility to graph f and the approximations, (c) determine whether**

**or**

**b the better approximation, and (d) state how the accuracy changes as you move farther from**

Calculus of a Single Variable

Ch. 2.1 - Tangent Line Describe how to find the slope of the...Ch. 2.1 - Notation List four notation alternatives to f(x).Ch. 2.1 - Derivative Describe how to find the derivative of...Ch. 2.1 - Continuity and Differentiability Describe the...Ch. 2.1 - Estimating Slope In Exercises 5 and 6, estimate...Ch. 2.1 - Estimating Slope In Exercises 5 and 6, estimate...Ch. 2.1 - Slopes of Secant Lines In Exercises 7 and 8, use...Ch. 2.1 - Slopes of Secant Lines In Exercises 7 and 8. use...Ch. 2.1 - Finding the Slope of a Tangent Line In Exercises...Ch. 2.1 - Finding the Slope of a Tangent Line In Exercises...

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Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Slope of a Tangent Line In Exercises 61 and 62,...Ch. 2.4 - Slope of a Tangent Line In Exercises 61 and 62,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Famous Curves In Exercises 79 and 80, find an...Ch. 2.4 - Famous Curves In Exercises 79 and 80, find an...Ch. 2.4 - Horizontal Tangent Line Determine the point(s) in...Ch. 2.4 - Horizontal Tangent Line Determine the point(s) at...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Evaluating a Second Derivative In Exercises 89-92,...Ch. 2.4 - Evaluating a Second Derivative In Exercises 89-92,...Ch. 2.4 - Evaluating a Second Derivative In Exercises 89-92,...Ch. 2.4 - Evaluating a Second Derivative In Exercises 89-92,...Ch. 2.4 - Identifying Graphs In Exercises 93 and 94, the...Ch. 2.4 - Identifying Graphs In Exercises 93 and 94, the...Ch. 2.4 - Describing Relationships Describe the relationship...Ch. 2.4 - Comparing Methods Consider the function...Ch. 2.4 - Think about It The table shows some values of the...Ch. 2.4 - Using Relationships Given that g(5)=3,g(5)=6,...Ch. 2.4 - Finding Derivatives In Exercises 99 and 100, the...Ch. 2.4 - Finding Derivatives In Exercises 99 and 100, the...Ch. 2.4 - Doppler Effect The frequency F of a fire truck...Ch. 2.4 - Harmonic Motion The displacement from equilibrium...Ch. 2.4 - Pendulum A 15-centimeter pendulum moves according...Ch. 2.4 - Wave Motion A buoy oscillates in simple harmonic...Ch. 2.4 - Modeling Data The normal daily maximum...Ch. 2.4 - HOW DO YOU SEE IT? The cost C (in dollars) of...Ch. 2.4 - Biology The number N of bacteria in a culture...Ch. 2.4 - Depreciation The value V of a machine t years...Ch. 2.4 - Finding a Pattern Consider the function f(x)=sinx,...Ch. 2.4 - Conjecture Let f be a differentiable function of...Ch. 2.4 - Think About It Let r(x)=f(g(x)) and s(x)=g(f(x)),...Ch. 2.4 - Using Trigonometric Functions (a) Find the...Ch. 2.4 - Even and Odd Functions (a) Show that the...Ch. 2.4 - Proof Let u be a differentiable function of x. Use...Ch. 2.4 - Using Absolute Value In Exercises 115-118. use the...Ch. 2.4 - Using Absolute Value In Exercises 115-118. use the...Ch. 2.4 - Using Absolute Value In Exercises 115-118. use the...Ch. 2.4 - Using Absolute Value In Exercises 115-118. use the...Ch. 2.4 - Linear and Quadratic Approximations The linear and...Ch. 2.4 - Linear and Quadratic Approximations The linear and...Ch. 2.4 - True or False? In Exercises 121-124. determine...Ch. 2.4 - True or False? In Exercises 121-124. determine...Ch. 2.4 - True or False? In Exercises 121-124. determine...Ch. 2.4 - True or False? In Exercises 121-124. determine...Ch. 2.4 - PUTNAM EXAM CHALLENGE Let...Ch. 2.4 - PUTNAM EXAM CHALLENGE Let k be a fixed positive...Ch. 2.5 - CONCEPT CHECK Explicit and Implicit Functions...Ch. 2.5 - CONCEPT CHECK Implicit Differentiation In your own...Ch. 2.5 - CONCEPT CHECK Implicit Differentiation Explain...Ch. 2.5 - Chain Rule How is the Chain Rule applied when...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding Derivatives Implicitly and Explicitly In...Ch. 2.5 - Finding Derivatives Implicitly and Explicitly In...Ch. 2.5 - Finding Derivatives Implicitly and Explicitly In...Ch. 2.5 - Finding Derivatives Implicitly and Explicitly In...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 2.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 2.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 2.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 2.5 - Famous Curves In Exercises 43-48, find an equation...Ch. 2.5 - Famous Curves In Exercises 43-48, find an equation...Ch. 2.5 - Famous Curves In Exercises 37-42, find an equation...Ch. 2.5 - Famous Curves In Exercises 37-42, find an equation...Ch. 2.5 - Famous Curves In Exercises 43-48, find an equation...Ch. 2.5 - Famous Curves In Exercises 43-48, find an equation...Ch. 2.5 - EXPLORING CONCEPTS Implicit and Explicit Forms...Ch. 2.5 - EXPLORING CONCEPTS Think About ItExplain why the...Ch. 2.5 - Ellipse (a) Use implicit differentiation to find...Ch. 2.5 - Hyperbola (a) Use implicit differentiation to find...Ch. 2.5 - Determining a Differentiable Function In Exercises...Ch. 2.5 - Determining a Differentiable Function In Exercises...Ch. 2.5 - Finding a Second Derivative In Exercises 49-54....Ch. 2.5 - Finding a Second Derivative In Exercises 49-54....Ch. 2.5 - Finding a Second Derivative In Exercises 49-54,...Ch. 2.5 - Finding a Second Derivative In Exercises, find...Ch. 2.5 - Finding a Second Derivative In Exercises, 49-54....Ch. 2.5 - Finding a Second Derivative In Exercises 49-54,...Ch. 2.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.5 - Tangent Lines and Normal Lines In Exercises 63 and...Ch. 2.5 - Tangent Lines and Normal Lines In Exercises 57 and...Ch. 2.5 - Normal Lines Show that the normal line at any...Ch. 2.5 - Circles Two circles of radius 4 are tangent to the...Ch. 2.5 - Vertical and Horizontal Tangent Lines In Exercises...Ch. 2.5 - Vertical and Horizontal Tangent Lines In Exercises...Ch. 2.5 - Orthogonal Trajectories In Exercises 63-66, use a...Ch. 2.5 - Orthogonal Trajectories In Exercises 63-66. use a...Ch. 2.5 - Orthogonal Trajectories In Exercises 63-66, use a...Ch. 2.5 - Orthogonal Trajectories In Exercises 81-84, use a...Ch. 2.5 - Orthogonal Trajectories In Exercises 85 and 86,...Ch. 2.5 - Orthogonal Trajectories In Exercises 85 and 86,...Ch. 2.5 - Orthogonal Trajectories The figure below shows the...Ch. 2.5 - HOW DO YOU SEE IT? Use the graph to answer the...Ch. 2.5 - Finding Equations of Tangent Lines Consider the...Ch. 2.5 - Tangent Lines and Intercepts Let l. be any tangent...Ch. 2.5 - Proof (a) Prove (Theorem 3.3) that d/dxxn=nxn1 for...Ch. 2.5 - Slope Find all points on the circle x2+y2=100...Ch. 2.5 - Tangent Lines Find equations of both tangent lines...Ch. 2.5 - Normals to a Parabola The graph shows die normal...Ch. 2.5 - Normal Lines (a) Find an equation of the normal...Ch. 2.6 - Related-Rate Equation What is a related-rate...Ch. 2.6 - Related Rates In your own words, state the...Ch. 2.6 - Using Related Rates In Exercises 3-6, assume that...Ch. 2.6 - Using Related Rates In Exercises 3-6, assume that...Ch. 2.6 - Using Related Rates In Exercises 3-6, assume that...Ch. 2.6 - Using Related Rates In Exercises 3-6, assume that...Ch. 2.6 - Moving Point In Exercises 7-10, a point is moving...Ch. 2.6 - Moving Point In Exercises 7-10, a point is moving...Ch. 2.6 - Moving Point In Exercises 710, a point is moving...Ch. 2.6 - Moving Point In Exercises 7-10, a point is moving...Ch. 2.6 - Area The radius r of a circle is increasing at a...Ch. 2.6 - Area The length .v of each side of an equilateral...Ch. 2.6 - Volume The radius r of a sphere is increasing at a...Ch. 2.6 - Radius A spherical balloon is inflated with gas at...Ch. 2.6 - Volume All edges of a cube are expanding at a rate...Ch. 2.6 - Surface Area All edges of a cube are expanding at...Ch. 2.6 - Height At a sand and gravel plant, sand is falling...Ch. 2.6 - Height The volume of oil in a cylindrical...Ch. 2.6 - Depth A swimming pool is 12 meters long, 6 meters...Ch. 2.6 - Depth A trough is 12 feet long and 3 feet across...Ch. 2.6 - Moving Ladder A ladder 25 feet long is leaning...Ch. 2.6 - Construction A construction worker pulls a...Ch. 2.6 - Construction A winch at the top of a 12-meter...Ch. 2.6 - Boating A boat is pulled into a dock by means of a...Ch. 2.6 - Air Traffic Control An air traffic controller...Ch. 2.6 - Air Traffic Control An airplane is flying at an...Ch. 2.6 - Sports A baseball diamond has the shape of a...Ch. 2.6 - Sports For (be baseball diamond in Exercise 27,...Ch. 2.6 - Shadow Length A man 6 feet tall walks at a rate of...Ch. 2.6 - Shadow Length Repeal Exercise 29 for a man 6 feet...Ch. 2.6 - Machine Design The endpoints of a movable rod of...Ch. 2.6 - Machine Design Repeat Exercise 31 for a position...Ch. 2.6 - Evaporation As a spherical raindrop falls, it...Ch. 2.6 - HOW DO YOU SEE IT? Using the graph of f, (a)...Ch. 2.6 - Think About It Describe the relationship between...Ch. 2.6 - EXPLORING CONCEPTS Volume Let V be the volume of a...Ch. 2.6 - Electricity The combined electrical resistance R...Ch. 2.6 - Electrical Circuit The voltage V in volts of an...Ch. 2.6 - Flight Control An airplane is flying in still air...Ch. 2.6 - Angle of Elevation A balloon rises at a rate of 4...Ch. 2.6 - Angle of Elevation A fish is reeled in at a rate...Ch. 2.6 - Angle of Elevation An airplane flies at an...Ch. 2.6 - Linear vs. Angular Speed A patrol car is parked 50...Ch. 2.6 - Linear vs. Angular Speed A wheel of radius 30...Ch. 2.6 - Area The included angle of the two sides of...Ch. 2.6 - Security Camera A security camera is centered 50...Ch. 2.6 - Modeling Data The table shows the numbers (in...Ch. 2.6 - Moving Shadow A ball is dropped from a height of...Ch. 2.6 - Acceleration In Exercises 49 and 50, find the...Ch. 2.6 - Acceleration In Exercises 49 and 50, find the...Ch. 2 - Finding the Derivative by the Limit Process In...Ch. 2 - Finding the Derivative by the Limit Process In...Ch. 2 - Finding the Derivative by the Limit Process In...Ch. 2 - Finding the Derivative by the Limit Process In...Ch. 2 - Using the Alternative Form of the Derivative In...Ch. 2 - Using the Alternative Form of the Derivative In...Ch. 2 - Determining Differentiability In Exercises 7 and...Ch. 2 - Determining Differentiability In Exercises 7 and...Ch. 2 - Finding a Derivative In Exercises 9-20, use the...Ch. 2 - Finding a Derivative In Exercises 9-20, use the...Ch. 2 - Finding a Derivative In Exercises 9-20, use the...Ch. 2 - Finding a Derivative In Exercises 9-20, use the...Ch. 2 - Finding a Derivative In Exercises 9-20, use (he...Ch. 2 - Finding a Derivative In Exercises 9-20, use the...Ch. 2 - Finding a Derivative In Exercises 9-20, use the...Ch. 2 - Finding a Derivative In Exercises 9-20, use the...Ch. 2 - Finding a Derivative In Exercises 9-20, use the...Ch. 2 - Finding a Derivative In Exercises 9-20, use the...Ch. 2 - Finding a Derivative In Exercises 920, use the...Ch. 2 - 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Finding a Derivative In Exercises 29-40, use the...Ch. 2 - Finding a Derivative In Exercises 29-40, use the...Ch. 2 - Finding a Derivative In Exercises 29-40, use the...Ch. 2 - Finding a Derivative In Exercises 29-40, use the...Ch. 2 - Finding an Equation of a Tangent Line In Exercises...Ch. 2 - Finding an Equation of a Tangent Line In Exercises...Ch. 2 - Finding an Equation of a Tangent Line In Exercises...Ch. 2 - Finding an Equation of a Tangent Line In Exercises...Ch. 2 - Finding a Second Derivative In Exercises 45-52,...Ch. 2 - Finding a Second Derivative In Exercises 4552,...Ch. 2 - Finding a Second Derivative In Exercises 4552,...Ch. 2 - Finding a Second Derivative In Exercises 4552,...Ch. 2 - Finding a Second Derivative In Exercises 4552,...Ch. 2 - Finding a Second Derivative In Exercises 4552,...Ch. 2 - Finding a Second Derivative In Exercises 4552,...Ch. 2 - Finding a Second Derivative In Exercises 4552,...Ch. 2 - Acceleration The velocity of an object is...Ch. 2 - Acceleration The velocity of an automobile...Ch. 2 - 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Moving Shadow A sandbag is dropped from a balloon...Ch. 2 - Finding Equations of Circles Consider the graph of...Ch. 2 - Finding Equations of Tangent Lines Graph the two...Ch. 2 - Finding a Polynomial Find a third-degree...Ch. 2 - Finding a Function Find a function of the form...Ch. 2 - Tangent Lines and Normal Lines (a) Find an...Ch. 2 - Finding Polynomials (a) Find the polynomial...Ch. 2 - Famous Curve The graph of the eight curve...Ch. 2 - Famous Curve The graph of the pear-shaped quartic...Ch. 2 - Shadow Length A man 6 feet tall walks at a rate of...Ch. 2 - Moving Point A particle is moving along the graph...Ch. 2 - Projectile Motion An astronaut standing on the...Ch. 2 - Proof Let E be a function satisfying E(0)=E(0)=1....Ch. 2 - Proof Let L be a differentiable function for all...Ch. 2 - Radians and Degrees The fundamental limit...Ch. 2 - Acceleration and Jerk If a is the acceleration of...

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In the following exercises, evaluate the indefinite integral f(x)dx with constant C = 0 using u-substitution. T...

Calculus Volume 2

Compute x and x for each of the values of x in 1-4. -14.00001

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14. Morningstar publishes ratings data on 1208 company stocks (Morningstar website, October 24, 2012). A sample...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

Rewriting Fractions In Exercises 29-34, show that the two expressions are equivalent. x23/10y24/5=10x235y24

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For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of . 117....

Calculus Volume 1

Salaries for teachers In a particular elementary school district are normally distributed with a mean of 54-1,0...

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Harris-Benedict Formula Your basal metabolic rate is the amount of energy in calories your body needs to functi...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

3. Refer to exercise 2.
what hypotheses are implied in this problem?
At the α = .05 level of significance, can ...

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In Problems 16 write the given linear system in matrix form. 6. dxdt=3x+4y+etsin2tdydt=5x+9z+4etcos2tdzdt=y+6ze...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)