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Calc 1 4.2

 

<== Calculus 1
Number Question Answer
See the work
1. Consider the function f(x) = 2-4x^2 on the interval
[ -4 , 8 ]. Find the average or mean slope of the function on this interval





By the Mean Value Theorem, we know there exists a c in the open interval (-4, 8) such that f'( c) is equal to this mean slope. Find the one c that works for this problem.
Mean slope= -16

c=2
The work

Point Cost: 2
2. Consider the function:
f(x) = rac {1} {x}
on the interval [ 1 , 11 ].

Find the average or mean slope of the function on this interval.

The Mean Value Theorem tells us that there exists a c in the open interval ( 1 , 11 ) such that f'( c) is equal to this mean slope. What is the only c that works.
Mean slope: -1/11

c= 3.317
The work

Point Cost: 2
3. Consider the function

f(x) = 3 x^3+ 4 x^2+ 2 x - 3
Find the average slope of this function on the interval ( -3 , 0 ).
By the Mean Value Theorem, we know there exists a c in the open interval ( -3, 0 ) such that f'(c) is equal to this mean slope. Find the value of c in the interval which works
Average slope: 17

c= -1.81
The work

Point Cost: 2
4. Consider the function f(x) = 4 \sqrt { x} + 2 on the interval [ 3 , 6 ]. Find the average or mean slope of the function on this interval.

By the Mean Value Theorem, we know there exists a c in the open interval ( 3 , 6 ) such that f'( c) is equal to this mean slope. For this problem, there is only one c that works. Find it.
Mean slope: 0.957

c= 4.37
The work

Point Cost: 2
5. Consider the function f(x) = 2 x^3 - 9 x^2 - 108 x + 7
on the interval [ -5 , 8 ].
Find the average or mean slope of the function on this interval.

By the Mean Value Theorem, we know there exists a c in the open interval ( -5 , 8 ) such that f'( c) is equal to this mean slope. For this problem, there are two values of c that work. The smaller one is

and the larger one is
Mean slope: -37

smaller c= -2.253

Larger c= 5.253
The work

Point Cost: 2
6. Consider the function f(x) = 3  x^3 - 4 x on the interval [ -4 , 4 ].
Find the average or mean slope of the function on this interval.

By the Mean Value Theorem, we know there exists at least one c in the open interval ( -4 , 4 ) such that f'( c) is equal to this mean slope.
For this problem, there are two values of c that work.
The smaller one is

and the larger one is

Mean slope: 44

Smaller c= -2.31

Larger c= 2.31
The work

Point Cost: 2
7. Consider the function f(x) = x^2-4x+9 on the interval [ 0 , 4 ]. Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval.
f(x) is on [0,4]
f(x) is on (0,4)
and f(0)=f(4)= .

Then by Rolle's theorem, there exists a c such that c.
c=
continuous for [0,4]

differentiable for (0,4)

f(0)=f(4)= 9

c= 2
The work

Point Cost: 2
8. Consider the function f(x) = x^3-3x^2+2x-12 on the interval [ 0 , 2 ]. Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval:

f(x) is on [0,2]
f(x) is on (0,2)
and f(0)=f(2)= .

Then by Rolle's theorem, there exists a c such that f'(c)=0.
Find the values c that satisfy the conclusion of Rolle's theorem.

c_1= and c_2= with c_1<c_2.
[0,2] is continuous

(0,2) is differentiable

f(0)=f(2)= -12

c_1=(3-sqrt(3))/3

c_1=(3+sqrt(3))/3
The work

Point Cost: 3
9. Suppose f(x) is continuous on [4,5] and -5 \le f'(x)\le 4 for all x in (4,5). Use the Mean Value Theorem to estimate f(5)-f(4).

Answer:
\le f(5)-f(4) \le
-5 \le f(5)-f(4) \le 4
The work

Point Cost: 1
10. Let f(x)=4 \sin(x).

a.) |f'(x)| \le
b.) By the Mean Value Theorem,
|f(a)-f(b)|\le |a-b| for all a and b.
|f'(x)| \le 4

|f(a)-f(b)|\le 4
The work

Point Cost: 1