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Solutions to various integration and differentiation problems for the mth132 course. It includes finding the definite integrals of given functions, evaluating the integrals at specific intervals, and determining the intervals where the functions are increasing or concave downward.
What you will learn
Typology: Lecture notes
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Question 1. Let g(x) =
∫ (^) x
0
f (t) dt, where f is the function whose graph is shown.
(a) Evaluate g(0), g(1), g(2), g(3), and g(6).
t
y
1
1
(^0 )
f
(b) On what interval is g increasing?
(c) Where does g have a maximum value?
(d) Sketch a rough graph of g.
x
y
1 2 3 4 5 6 7
Question 2. Evaluate the integral:
(a)
1
(8x 3
(b)
0
u − 2 u 2
u
du
(c)
0
|t
2 − 4 | dt
(d)
0
(1 + 2x)
3 dx
Question 4. If f (x) =
∫ (^) x
0
(1 − t
2 ) cos
2 t dt, on what interval is f increasing?
Question 5. On what interval is the curve y =
∫ (^) x
0
t
2
t^2 + t + 2
dt
concave downward?
