Calculus AB AP
Worksheet: Fundamental Theorem of Calculus I

f  b   f  a    f '  x  dx , and f  b   f  a    f '  x  dx
b
b
a
a
Part I: Non-Calculator
1. If f  x   ln 1  sin x  , find


2
0
f '  x  dx .
2. Let f be a continuous and differentiable function such that f '  x   4  x 2 and f  0   3 . Find the
value of f  2  .
3. Let f be a differentiable function. The derivative of f , f ' , is shown in
the figure on the right. If f  0   4 , find the value of f  2  .
4. The graph of f ' , the derivative of a function f , on the
interval x   0,8 , is shown in the figure. Given that
f  4   5 , find the values of f  8  , f  6  , f  2  , and
f  0 .
5. The graph of f ' , the derivative of a function f , is
shown in the figure. In the figure, the area of the region
A 1 is 3, and the area of the region A 2 is 8. Given that
f  1  6 , find the values of f  4  and f  4  .
6. The graph of f ' , the derivative of a function f , on the interval
x   1, 4 , is shown in the figure. Given that f 1  2 , find
the values of f  1 , f  2  , and f  4  .
Part II: Calculator
7. Let f '  x   ln  e x  2  , and f 1  4 . Find the values of f  5  and f  5  .
8. The graph of f ' , the derivative of a function f ,
on the interval x   0,8 , is shown in the figure.
Rank the following values from the least to the
greatest: f  0  , f  2  , f  5  , and f  8  .
9. Let f be a function defined on x  1, 4 . If f '  x   ln  x  sin x  , and f 1  4 ,
(a) Find f  4  .
(b) Find the x-coordinate of the absolute minimum of f on the interval x  1, 4 .
(c) Find the absolute minimum value of f on the interval x  1, 4 .
1
, and f 1  2 . Find the values of f  0  , f  2  and f  4  . If any of the
sin x  cos x
values are undefined, explain why.
10. Let f '  x  
 e x  e x 
11. Let f be an everywhere differentiable function with derivative f '  x   ln 
 , and
 2 
3
f  1   . Find the values of f  4  , f  0  and f  4  .
2
12. Let f be a differentiable function with derivative f '  x   arctan  x sin x  .
(a) Find the average rate of change of f on the interval x  1, 4 .
(b) Find the value(s) of c on the open interval 1,4  that satisfies the conclusion of the Mean Value
Theorem of Differential Calculus.