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10th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781285057095

Chapter 2.1, Problem 27E

Textbook Problem

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Finding an Equation of a Tangent Line In Exercises 29-36, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results.

**(a)**

To determine

**To calculate:** Equation of the tangent line at the given point.

**Given:**

**Formula Used:**

**Calculation:**

To find the slope of the tangent line we must find the derivative of the function at point (1, 4)

To find the slope of the ta

**(b)**

To determine

**To Graph:** The function using the graphing utility.

**(c)**

To determine

**To Graph:** A tangent using graphing utility.

Calculus

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Using the Constant Multiple Rule In Exercises...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 2538, find the...Ch. 2.3 - Finding a Derivative In Exercises 2538, find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 2538, find the...Ch. 2.3 - Finding a Derivative In Exercises 2538, find the...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - 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Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Higher-Order Derivative In Exercises...Ch. 2.3 - Finding a Higher-Order Derivative In Exercises...Ch. 2.3 - Finding a Higher-Order Derivative In Exercises...Ch. 2.3 - Finding a Higher-Order Derivative In Exercises...Ch. 2.3 - Using Relationships In Exercises 105-108. use the...Ch. 2.3 - Using Relationships In Exercises 105-108. use the...Ch. 2.3 - Using Relationships In Exercises 105-108, use the...Ch. 2.3 - Using Relationships In Exercises 105-108. use the...Ch. 2.3 - Sketching a Graph Sketch the graph of a...Ch. 2.3 - Sketching a Graph Sketch the graph of a...Ch. 2.3 - Identifying Graphs In Exercises 111 and 112, the...Ch. 2.3 - Identifying Graphs In Exercises 111 and 112, the...Ch. 2.3 - Sketching Graphs In Exercises 113 and 114, the...Ch. 2.3 - Sketching Graphs In Exercises 111114, the graph of...Ch. 2.3 - Sketching Graphs In Exercises 113 and 114, the...Ch. 2.3 - Sketching Graphs In Exercises 111114, the graph of...Ch. 2.3 - Acceleration The velocity of an object is...Ch. 2.3 - Acceleration The velocity of an automobile...Ch. 2.3 - Stopping Distance A car is traveling at a rate of...Ch. 2.3 - HOW DO YOU SEE IT? The figure shows the graphs of...Ch. 2.3 - Finding a Pattern In Exercises 121 and 122,...Ch. 2.3 - Finding a Pattern In Exercises 121 and 122....Ch. 2.3 - Finding a Pattern Consider the function...Ch. 2.3 - Finding a Pattern Develop a general rule for the...Ch. 2.3 - Finding a Pattern In Exercises 125 and 126, find...Ch. 2.3 - Finding a Pattern In Exercises 125 and 126, find...Ch. 2.3 - Differential Equations In Exercises 127130, verify...Ch. 2.3 - Differential Equations In Exercises 127-130,...Ch. 2.3 - Differential Equations In Exercises 127-130,...Ch. 2.3 - Differential Equations In Exercises 127-130,...Ch. 2.3 - True or False? In Exercises 131-136, determine...Ch. 2.3 - True or False? In Exercises 131-136. determine...Ch. 2.3 - True or False? In Exercises 131-136. determine...Ch. 2.3 - True or False? In Exercises 129134, determine...Ch. 2.3 - True or False? In Exercises 131-136, determine...Ch. 2.3 - True or False? In Exercises 129134, determine...Ch. 2.3 - Absolute Value Find the derivative of f(x) = x|x|....Ch. 2.3 - Proof Use the Product Rule twice to prove that if...Ch. 2.3 - Think About It Let f and g be functions whose...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Finding a Derivative In Exercises 734, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 734, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 734, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 734, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 734, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 734, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a Derivative In Exercises 734, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 734, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...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 41 and 42,...Ch. 2.4 - Slope of a Tangent Line In Exercises 41 and 42,...Ch. 2.4 - Finding a Derivative of a Trigonometric FunctionIn...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 In Exercises 4364, find the...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 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 In Exercises 4364, find the...Ch. 2.4 - Finding a Derivative In Exercises 4364, find the...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 In Exercises 4364, find the...Ch. 2.4 - Finding a Derivative In Exercises 4364, find the...Ch. 2.4 - Finding a Derivative In Exercises 43 64, find the...Ch. 2.4 - Finding a Derivative In Exercises 43 64, find the...Ch. 2.4 - Finding a Derivative In Exercises 43 64, find the...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...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 - Evaluating a Derivative In Exercises 6572, find...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Evaluating a Derivative In Exercises 6572, find...Ch. 2.4 - Evaluating a Derivative In Exercises 6572, find...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 8590,...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 - WRITING ABOUT CONCEPTS Identifying Graphs In...Ch. 2.4 - WRITING ABOUT CONCEPTS Identifying Graphs In...Ch. 2.4 - Describing Relationships Describe the relationship...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 125128, determine...Ch. 2.4 - True or False? In Exercises 125128, determine...Ch. 2.4 - True or False? In Exercises 121-124. determine...Ch. 2.4 - True or False? 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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 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 - WRITING ABOUT CONCEPTS Related Rates Consider the...Ch. 2.6 - Area The radius r of a circle is increasing at a...Ch. 2.6 - 12. 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Volume 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 29-42, use the...Ch. 2 - Finding a Derivative In Exercises 29-40, use the...Ch. 2 - Finding a Derivative In Exercises 2940, 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 29-42, use the...Ch. 2 - Finding a Derivative In Exercises 2940, use the...Ch. 2 - Finding a Derivative In Exercises 29-40, use the...Ch. 2 - Finding a Derivative In Exercises 2940, 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...Ch. 2 - Finding a Second Derivative In Exercises 45-50,...Ch. 2 - Finding a Second Derivative In Exercises 45-50,...Ch. 2 - Finding a Second Derivative In Exercises 45-50,...Ch. 2 - Finding a Second Derivative In Exercises 45-50,...Ch. 2 - Finding a Second Derivative In Exercises 45-50,...Ch. 2 - Finding a Second Derivative In Exercises 45-50,...Ch. 2 - Acceleration The velocity of an object in meters...Ch. 2 - Acceleration The velocity of an automobile...Ch. 2 - Find a Derivative In Exercise 53- 64, find the...Ch. 2 - Find a Derivative In Exercise 53- 64, find the...Ch. 2 - Finding a Derivative In Exercises 5364, find the...Ch. 2 - Finding a Derivative In Exercises 53-64, find the...Ch. 2 - Finding a Derivative In Exercises 53-64, find the...Ch. 2 - Finding a Derivative In Exercises 5364, find the...Ch. 2 - Finding a Derivative In Exercises 53-64, find the...Ch. 2 - Finding a Derivative In Exercises 53-64, find the...Ch. 2 - Finding a Derivative In Exercises 53-64, find the...Ch. 2 - Finding a Derivative In Exercises 53-64, find the...Ch. 2 - Finding a Derivative In Exercises 5364, find the...Ch. 2 - Finding a Derivative In Exercises 53-64, find the...Ch. 2 - Finding a Derivative In Exercises 65-70, find and...Ch. 2 - Finding a Derivative InExercises 65-70, find and...Ch. 2 - Finding a Derivative In Exercises 65-70, find and...Ch. 2 - Finding a Derivative InExercises 65-70, find and...Ch. 2 - Finding a Derivative InExercises 65-70, find and...Ch. 2 - Finding a Derivative In Exercises 65-70, find and...Ch. 2 - Finding the Second Derivative InExercises 71-74,...Ch. 2 - Finding a Second Derivative In Exercises 71-74,...Ch. 2 - Finding a Second Derivative In Exercises 71-74,...Ch. 2 - Finding a Second Derivative In Exercises 7174,...Ch. 2 - 75. Refrigeration The temperature T (in degrees...Ch. 2 - Harmonic Motion The displacement from equilibrium...Ch. 2 - Finding a Derivative In Exercises 77-82, find...Ch. 2 - Finding a Derivative In Exercises 77-82, find...Ch. 2 - Finding a Derivative In Exercises 77-82, find...Ch. 2 - Finding a Derivative In Exercises 77-82, find...Ch. 2 - Finding a Derivative In Exercises 77-82, find...Ch. 2 - Finding a Derivative In Exercises 77-82, find...Ch. 2 - Tangent Lines and Normal Lines In Exercises 83 and...Ch. 2 - Tangent Lines and Normal Lines In Exercises 83-84,...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 rotating 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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College Algebra (MindTap Course List)

Miles per Gallon. A consumer research group is interested in testing an automobile manufacturers claim that a n...

Essentials Of Statistics For Business & Economics

For each exercise, functions and cofunctions of two angles are given. Which of the functions or cofunctios of t...

Mathematics For Machine Technology

Define extraneous variable and explain how extraneous variables can become confounding variables.

Research Methods for the Behavioral Sciences (MindTap Course List)