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Chapter 2: Q. 3 (page 236)

If f is a differentiable function, then the values x=cat which the sign of the second derivative f''xchanges are the locations of the inflection points of f. Use this information to find the inflection points of the functionfx=sinx. Illustrate your answer on a graph ofy=sinx.

Short Answer

Expert verified

The local extrema of the given function is at x=nπ. The graph of the given function is given below,

Step by step solution

01

Step 1. Given information.

Consider the given question,

The function isfx=sinx.

02

Step 2. Find the derivatives of the given function.

Derivative of the given function,

f'x=cosx

For critical points, put f'x=0. Then,

0=cosxx=...,-3π2,-π2,0,π2,3π2,...x=2n+1π2

Where, n is an integer.

The second derivate of the given function,

f''x=-sinx

When (2n-1)π<x<2nπ,n=0,±1,±2,…f''x=-sinx>0.

When 2nπ<x<2n+1π,n=0,±1,±2,...f''x=-sinx<0.

Therefore, the required graph isy=sinx.

03

Step 3. Plot the function.

On plotting the function is given below,

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Most popular questions from this chapter

Use the definition of the derivative to find ffor each function fin Exercises 39-54

f(x)=1x2-1

Think about what you did today and how far north you were from your house or dorm throughout the day. Sketch a graph that represents your distance north from your house or dorm over the course of the day, and explain how the graph reflects what you did today. Then sketch a graph of your velocity.

Stuart left his house at noon and walked north on Pine Street for 20minutes. At that point he realized he was late for an appointment at the dentist, whose office was located south of Stuart’s house on Pine Street; fearing he would be late, Stuart sprinted south on Pine Street, past his house, and on to the dentist’s office. When he got there, he found the office closed for lunch; he was 10minutes early for his 12:40appointment. Stuart waited at the office for 10minutes and then found out that his appointment was actually for the next day, so he walked back to his house. Sketch a graph that describes Stuart’s position over time. Then sketch a graph that describes Stuart’s velocity over time.

Velocity v(t) is the derivative of position s(t). It is also true that acceleration a(t) (the rate of change of velocity) is the derivative of velocity. If a race car’s position in miles t hours after the start of a race is given by the function s(t), what are the units of s(1.2)? What are the units and real-world interpretation of v(1.2)? What are the units and real-world interpretations of a(1.2)?

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

f'(x)=1-4x6;f(1)=3
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