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Chapter 4: Q. 4 (page 384)

State the Mean Value Theorem, and illustrate the theorem with a graph.

Short Answer

Expert verified

Thetangentlineatpointcisparalleltothesecantlineconnectingthe(a,f(a))and(b,f(b)).Slopeofsecantlineis:f'(c)=f(b)-f(a)b-a

Step by step solution

01

Step 1. Given information is:

Pointcin(a,b)suchthatf'(c)=f(b)-f(a)b-a

02

Step 2. Graph and Interpretation

Geometrically,itgivestherelationbetweenslopeofthesecantlineandslopeoftangentline.

Thetangentlineatpointcisparalleltothesecantlineconnectingthe(a,f(a))and(b,f(b)).Slopeofsecantlineis:f'(c)=f(b)-f(a)b-a

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Most popular questions from this chapter

Given a simple proof that if n is a positive integer and c is any real number, then鈭�k=1nc=cn

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

鈭�04(2x-3)2+5dx

Properties of addition: State the associative law for addition, the commutative law for addition, and the distributive law for multiplication over addition of real numbers. (You may have to think back to a previous algebra course.)

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 absolute area between the graph of f and the x-axis on [a, b] is equal to|鈭�abf(x)dx|.

(b) True or False: The area of the region between f(x) = x 鈭� 4 and g(x) = -x2on the interval [鈭�3, 3] is negative.

(c) True or False: The signed area between the graph of f on [a, b] is always less than or equal to the absolute area on the same interval.

(d) True or False: The area between any two graphs f and g on an interval [a, b] is given by 鈭�ab(f(x)-g(x))dx.

(e) True or False: The average value of the function f(x) = x2-3 on [2, 6] is

f(6)+f(2)2= 33+12= 17.

(f) True or False: The average value of the function f(x) = x2-3on [2, 6] is f(6)-f(2)4= 33-14= 8.

(g) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 2] and the average value of f on [2, 5].

(h) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 3] and the average value of f on [3, 5].

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value

limn鈫�鈭�鈭�k=1n(k2+k+1)
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