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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=n蟺. 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=...,-3蟺2,-蟺2,0,蟺2,3蟺2,...x=2n+1蟺2

Where, n is an integer.

The second derivate of the given function,

f''x=-sinx

When (2n-1)蟺<x<2n蟺,n=0,卤1,卤2,鈥�f''x=-sinx>0.

When 2n蟺<x<2n+1蟺,n=0,卤1,卤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

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=xx+1-2

Write down a rule for differentiating a composition f(u(v(w(x))))of four functions

(a) in 鈥減rime鈥� notation and

(b) in Leibniz notation.

Every morning Linda takes a thirty-minute jog in Central Park. Suppose her distance s in feet from the oak tree on the north side of the park tminutes after she begins her jog is given by the function s(t)shown that follows at the left, and suppose she jogs on a straight path leading into the park from the oak tree.

(a) What was the average rate of change of Linda鈥檚 distance from the oak tree over the entire thirty-minute jog? What does this mean in real-world terms?

(b) On which ten-minute interval was the average rate of change of Linda鈥檚 distance from the oak tree the greatest: the first 10minutes, the second 10minutes, or the last10minutes?

(c) Use the graph of s(t)to estimate Linda鈥檚 average velocity during the 5-minute interval fromt=5tot=10. What does the sign of this average velocity tell you in real-world terms?

(d) Approximate the times at which Linda鈥檚 (instantaneous) velocity was equal to zero. What is the physical significance of these times?

(e) Approximate the time intervals during Linda鈥檚 jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?

Use the limit you just found to calculate the exact slope of the tangent line to f(x)=x2at x=4. Obviously you should get the same final answer as you did earlier.

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=1+x23x2-4x+1
See all solutions

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