Download Average Value of a Function, The 2 nd Fundamental Theorem of

Avon High School
AP Calculus AB
Name ___________________________________________
HW 4.4 B – Average Value of a Function, The 2nd Fundamental Theorem of Calculus
& The Accumulation Function
Period _____ Score ______ / 10
1. Find the average value of the function f on the given interval. Do not use your TI-Nspire.
a. f ( x)  x2  2 x 0,3
 
b. f ( x)  cos x 0, 
 2
c. f ( x)  x 1,9
c.
f ( x) 
1
x2
1,5
2. Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the
given interval. Use your TI-Nspire on c. and d.
a. f ( x)  4  x2 , 0, 2
b. f ( x)  4 x  x2 , 0,3
c. f ( x)  x3  x  1, 0, 2
 
d. f ( x)  1  tan 2 x, 0, 
 4
3. Find the number b such that the average value of f ( x)  2  7 x  x3 on the interval 0,b is equal to 3.
Show your setup and integration work. Only use your TI-Nspire to solve for b.
4. In a certain city, the temperature (in °F) t hours after 9 A.M. is approximated by the function
T (t )  50  14sin
t
12
. Find the average temperature during the period 9 A.M. to 9 P.M.
Use your TI-Nspire to integrate the result.
x
2
dt . Take the integral, then
t2
1
5.) Show that the Second Fundamental Theorem of Calculus holds for F ( x)  
take the derivative.
6.) Use the Second Fundamental Theorem of Calculus to find the derivatives of the following functions.
x
x
a. f ( x)    t 2  1 dt
20
b. g ( x) 

1
1
x
x2
1
dt
c. g ( x)  
4
 1 t
t 3  1 dt
d. f ( x)   cos(t 2 ) dt
4
x
7. Find the interval on which the curve y   (t 3  t 2  1) dt is concave up. Justify your answer.
0
x
8.) Let F ( x)   f (t ) dt where the graph of f ( x) is above. The graph consists of lines and a quarter circle.
0
a. Complete the chart.
x
F ( x)
F ( x)
0
1
2
3
4
5
7
8
b. On what subintervals of 0,8 is F increasing? Decreasing? Justify your answer.
c. Where in the interval 0,8 does F achieve
its minimum value? What is the minimum
value? Justify your answer.
d. Where in the interval 0,8 does F achieve
its maximum value? What is the maximum
value? Justify your answer.
e. Find the concavity of F and any inflection
points. Justify your answer.
f. Sketch a rough graph of F ( x) .
x
9.) Let F ( x)   f (t ) dt where the graph of f ( x) is above. The graph consists of lines and a quarter circle.
0
a. Complete the chart. You will have to approximate values at x  1 and x  1.
x
F ( x)
F ( x)
4
3
2
1
0
1
2
3
4
5
6
7
8
b. On what subintervals of  4,8 is F increasing? Decreasing? Justify your answer.
c. Where in the interval  4,8 does F achieve
its minimum value? What is the minimum
value? Justify your answer.
d. Where in the interval  4,8 does F achieve
its maximum value? What is the maximum
value? Justify your answer.
e. Find the concavity of F and any inflection
points. Justify your answer.
f. Sketch a rough graph of F ( x) .
10.) Buffon’s Needle Experiment A horizontal plane is ruled with parallel lines 2 inches apart. A two-inch
needle is tossed randomly onto the plane. The probability that the needle will touch a line is

P
2

2
 sin  d
0
where  is the acute angle between the needle and any one of the parallel lines. Find this probability.