Calculus2
The Fundamental Theorem of Calculus (Part 1)
The Area Function
1. Let g(x) = ∫ x 0 f(t) dt, where f is the function whose graph is shown in Figure 1.
a) Reproduce and fill out the table.
x 1 2 3 4 5 6
g(x)
b) State the interval(s) over which g is (i) increasing, and (ii) decreasing.
t
y
1 2 3 4 5 6 −1
1
2
3
y = f(t)
Figure 1
t
y
1 3 5
−3
−2
−1
1
2 y = f(t)
Figure 2
2. Let g(x) = ∫ x 0 f(t) dt, where f is the function whose graph is shown in Figure 2.
a) State the interval(s) over which g is (i) increasing, and (ii) decreasing.
b) State the value(s) of x at which g(x) is (i) maximum, and (ii) minimum.
3. Let g(x) = ∫ x 0 cos t dt.
a) Use the coordinate grid shown to estimate the values of g. Reproduce and fill out the table.
x 0 1 2 3 4 5 6 2π
g(x)
b) Use your data to sketch the graph of g.
t
y
1 3 5 6 −1
1 f(t) = cos t
1 of 2
The FTC (Part 1)
4. State the Fundamental Theorem of Calculus (Part 1), including conditions.
� 5–8 Use the FTC (Part 1) to find the derivative of the function.
5. g(x) =
∫ x 0
√ 1 + 2t dt
6. g(x) =
∫ x 1
ln t
t dt
7. h(x) =
∫ √x 1
e 1−t2
dt
8. h(x) =
∫ 1/x
π
arctan t dt
Solutions to Selected Problems
1. a) x 1 2 3 4 5 6
g(x) 1 2
0 −1 2
0 3 2
7
2
b) i. [0, 1] ∪ [3, 6]
ii. [1, 3]
2. a) i. [2, 4]
ii. [0, 2] ∪ [4, 6]
b) i. x = 4
ii. x = 6
3. a) x 0 1 2 3 4 5 6 2π
g(x) 0 0.8 0.9 0.1 -0.8 -0.9 -0.3 0
b) Graph
4. FTC (Part 1)
5. g′(x) = √ 1 + 2x
6. g′(x) = ln x
x
7. h′(x) = e1−x
2 √ x
8. h′(x) = − arctan 1
x
x2
2 of 2