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Chapter 10: Q. 40 (page 777)

For each function f and value x = c in Exercises 35鈥44, use a sequence of approximations to estimate f'(c). Illustrate your work with an appropriate sequence of graphs of secant lines.

role="math" localid="1648705352169" f(x)=ex,c=0

Short Answer

Expert verified

We have approximated the slope by using the concept of the secant line.

Step by step solution

01

Step 1. Given information.

We have to use a sequence of approximations to estimate f'(c)

f(x)=ex,c=0

02

Step 2. Use sequence of approximation

Let,

h=1,0.5,0.1,0.01

Consider the expressions,

f(1)鈭f(0)1鈭0=[e]鈭11=1.718f(0.5)鈭f(0)0.5鈭0=e0.5鈭10.5=1.297f(0.1)鈭f(0)1.1鈭0=e0.1鈭[1]0.1=1.051f(0.01)鈭f(0)0.01鈭0=e0.01鈭[1]0.01=1.005

The slope of tangent will be :

f鈥(0)=1

The graph is ;

03

Step 3. First secant graph

Take c=0 , c+h=1, then the corresponding values are:

f(0)=1,f(1)=2.718

The secant line can be drawn as:

04

Step 4. Second secant graph

Take c=0 and c+h = 0.1 then the corresponding values are :

f(0)=1,f(0.1)=1.1051

The secant graph is :

05

Step 5. Third secant graph

Take c=0 and c+h=0.01, then the corresponding values are :

f(0)=1,f(0.01)=1.0101

The secant graph is :

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Most popular questions from this chapter

Give an example of three nonzero vectors u, v and w in 鈩3 such that u脳v=u脳w but v鈮w. What geometric relationship must the three vectors have for this to happen?

Find PQ鈫

P=-1,5,4,Q=-1,6,4

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is k=1.

(b) True or False: 鈭k=0n鈥1k+1+鈭k=1n鈥k2is equal to 鈭k=0n鈥k3+k2+1k+1.

(c) True or False: 鈭k=1n鈥1k+1+鈭k=0n鈥k2is equal to 鈭k=1n鈥k3+k2+1k+1 .

(d) True or False: 鈭k=1n鈥1k+1鈭k=1n鈥k2 is equal to 鈭k=1n鈥k2k+1.

(e) True or False: 鈭k=0m鈥k+鈭k=mn鈥kis equal to鈭k=0n鈥k.

(f) True or False: 鈭k=0n鈥ak=鈭a0鈭an+鈭k=1n鈭1鈥ak.

(g) True or False: 鈭k=110鈥ak2=鈭k=110鈥ak2.

(h) True or False: 鈭k=1n鈥ex2=exex+12ex+16.

In Exercises 36鈥41 use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

(c) The area of the triangle determined by the points.

P(1,6),Q(0,鈭3),R(鈭5,4)

(Hint: Think of the xy-plane as part of 鈩3.)

In Exercises 22鈥29 compute the indicated quantities when u=(2,1,鈭3),v=(4,0,1),andw=(鈭2,6,5)

Find the area of the parallelogram determined by the vectors v and w.
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