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Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275361

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Chapter
Section
BuyFindarrow_forward

Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275361
Chapter 2.2, Problem 41E
Textbook Problem
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Finding a Derivative In Exercises 39-54, find the derivative of the function.

g ( t ) = t 2 4 t 3

To determine

To calculate: The derivative of the function g(t)=t24t3.

Explanation of Solution

Given: The function g(t)=t24t3

Calculation:

Use the power rule of differentiation which states that the function of the form f(x)=xn, n is a rational number, is differentiable, and is written as:

ddx[xn]=nxn1

The constant rule multiple states that if f is a differentiable function and c is a real number, then cf is also differentiable and ddx[cf(x)]=cf'(x).

Use the sum and difference rule of differentiation which states that the sum or difference of two differentiable functions is itself differentiable.

This can be written as:

ddx[f(x)+g(x)]=f'(x)+g'(x)

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Chapter 2 Solutions

Calculus of a Single Variable
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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? 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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 - Finding a Derivative In Exercises 920, use the...Ch. 2 - Finding the Slope of a Graph In Exercises 21-24,...Ch. 2 - Finding the Slope of a Graph In Exercises 21-24,...Ch. 2 - Finding the Slope of a Graph In Exercises 21-24,...Ch. 2 - Finding the Slope of a Graph In Exercises 21-24,...Ch. 2 - Vibrating String When a guitar string is plucked,...Ch. 2 - Surface Area The surface area of a cube with sides...Ch. 2 - Vertical Motion In Exercises 27 and 28, use the...Ch. 2 - Vertical Motion In Exercises 27 and 28, 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 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 a Derivative In Exercises 29-40, use the...Ch. 2 - 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Finding the Slope of a Graph In Exercises 67-72,...Ch. 2 - Finding a Second Derivative In Exercises 73-76,...Ch. 2 - Finding a Second Derivative In Exercises 73-76,...Ch. 2 - Finding a Second Derivative In Exercises 75-76,...Ch. 2 - Finding a Second Derivative In Exercises 75-76,...Ch. 2 - Refrigeration The temperature T (in degrees...Ch. 2 - Harmonic Motion The displacement from equilibrium...Ch. 2 - Finding a Derivative In Exercises 79-84, find...Ch. 2 - Finding a Derivative In Exercises 79-84, find...Ch. 2 - Finding a Derivative In Exercises 79-84, find...Ch. 2 - Finding a Derivative In Exercises 79-84, find...Ch. 2 - Finding a Derivative In Exercises 79-84, find...Ch. 2 - Finding a Derivative In Exercises 79-84, find...Ch. 2 - Tangent Lines and Normal Lines In Exercises 85 and...Ch. 2 - Tangent Lines and Normal Lines In Exercises 85 and...Ch. 2 - Rate of Change A point moves along the curve y=x...Ch. 2 - Surface Area All edges of a cube are expanding at...Ch. 2 - Linear vs. Angular Speed A routing beacon is...Ch. 2 - 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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