CALCULUS I: REVIEW SHEET FOR THE THIRD MIDTERM EXAM Problem #1. Use the Riemann sum with 8 segments and midpoints as sample points to compute the following: 1. Area of the shape enclosed by the x-axis, y-axis, the vertical line x = 1 2 and the curve y = e−x ; 2. The definite integral
1 2 −1 sin(x )
dx.
Problem #2. The function f (x) is given by its graph below. Find an overestimation and an underestimation of the following definite integrals. The exact number of segments (a.k.a “n”) is left up to you. a)
1 −1 f (x)
dx
2.5
b)
2 −2 f (x)
dx.
2
1.5
1
0.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
-0.5
-1
-1.5
-2
-2.5
Problem #3. Please compute the following definite integrals by interpreting them as “signed” areas. a) c)
2 √ −2 4 1 −1 |x|
− x2 dx
b) d)
10 3 −10 x 1 −1 |x
dx
− 1 dx
− 1| dx.
Problem #4. State and explain the Fundamental Theorem of Calculus (parts I and II). Then state the Net Change Theorem. 1
Problem #5. Find the functions f and g if it is known that: 1. f (x) = 2ex , f (0) = 0, f (0) = 1; g(0) = 0, g (0) = 1, g (0) = 0.
2. g (x) = sin(x),
Problem #6. Find the following indefinite integrals. You may need to use the substitution rule. a) c) e) g)
2e−x 5
dx
1 2x+1 )
b)
t+1 √ t
dt dx
2x ( x2 +1 + √ cos( x) √ x
dx
d) f) h)
8x+3 12x2
2
dx
e−u u du sin(ez + 1)ez dz.
esin y cos y dy
Problem #7. Find the first and the second derivatives of the following functions: a) f (x) =
0 x
sin(t2 ) dt
b) g(x) =
0
x
e−t dt.
2
Problem #8. Compute: a)
1 0 cos(2πt)
dt
b) d) f)
1 −1 (2u
+ 1)3 du dx
c) 1 e)
e t+1 t
dt
1 −x2 x 0 e 1 −2 |x|
1 2 0 (x
+ 1)2 dx
dx.
Problem #9. Particles P and Q are moving on a straight road. What can you say about the net change in their position over the time period 0 ≤ t ≤ 2 if it is known that: 1. The velocity of the particle P is described by v(t) = t2 − 3t + 2. 2. The acceleration of the particle Q is given by a(t) = 6t − 16 and the initial velocity of the particle Q was v(0) = 3. Problem #10. What is the area enclosed by the curve y = sin x and the x-axis, between x = 0 and x = π? 2