Math 1210
Practice for Midterm 3
February 28th, 2014
The following is a list of questions covering sections 3.6 through 4.1. The
questions in the midterm will be a subset of these, with different functions. I
will use this as a base for next Wednesday’s review session (April 2nd):
please try to think about as many exercises as possible so that you can ask
me questions on Wednesday.
Name:
UID:
Section 3.6: The intermediate value theorem for derivatives
1. Let f be a function which is differentiable over the open interval (a, b) and continuous
over the closed interval [a, b]. What does the intermediate value theorem for derivatives
say? If moreover f (a) = f (b), what does the theorem say?
2. Consider the function f (x) = x2/3 . Show that the conclusion of the mean value theorem
fails and explain why.
3. Consider the function f (x) = 2x3 − 9x2 + 1.
(i) Show that the equation f (x) = 0 has exactly one solution on the interval (−1, 0).
(ii) Show that the equation f (x) = 0 has exactly one solution on the interval (0, 1).
(iii) Show that the equation f (x) = 0 has exactly one solution on the interval (4, 5).
Section 3.7: Solving equations numerically
I won’t be asking anything about this section in the midterm.
Section 3.6: Antiderivatives
R 2 2
4. Compute the integral (z √+1)
dz. Hint: Expand the numerator first.
z
R
5. Compute the integral (t2 − 2 cos t) dt.
√
R
6. Compute the integral (5x2 + 1) 5x3 + 3x − 2 dx.
R
7. Compute the integral √ 3y2 dy.
2y +5
R
sin x(1 + cos x)4 dx.
√
R
9. Compute the integral sin x cos x 1 + sin2 x dx.
8. Compute the integral
Section 3.7: Introduction to differential equations
10. Solve the differential equation
dy
x + 3x2
=
dx
y2
and find the solution y(x) that satisfies y(0) = 6.
11. Solve the differential equation
dy
= −y 2 x(x2 + 2)4
dx
and find the solution y(x) that satisfies y(0) = 1.
Section 4.1: Introduction to area
P
12. Find a formula for nj=1 (j + 2)(j − 5).
13. Find the value of the following sums:
P7 (−1)k 2k
(i)
k=3 k+1
P6
kπ
(ii)
k=−1 k sin 2
14. Follow the steps below to show that if r is any real number different from 1, then
n
X
rk = r0 + r1 + . . . + rn =
k=0
(i) If S =
Pn
k=0
1 − rn+1
1−r
rk , what is rS.
(ii) Compute rS − S and solve for S.
P
k
(iii) Calculate the sum 10
k=1 2 .
15. Find the area of the region under the curve y = 2x + 2 over the interval [−1, 1]
(i) First sketch a graph of the function and find the area using the classical formula
for the area of a trapezoid.
(ii) Subdivide the interval [-1,1] into n equal subintervals, calculate the sum of the areas
of the corresponding n inscribed rectangles and let n → ∞.
16. Find the area of the region under the curve y = x2 over the interval [−2, 2]. To do this,
subdivide the interval [-2,2] into n equal subintervals, calculate the sum of the areas of
the corresponding n inscribed rectangles and let n → ∞.
17. Find the area of the region under the curve y = x3 + x over the interval [0, 1]. To do
this, subdivide the interval [0,1] into n equal subintervals, calculate the sum of the areas
of the corresponding n inscribed rectangles and let n → ∞.
Section 4.3: The first fundamental theorem of Calculus
For the following questions, remember that if f is a continuous function on the closed
interval [a, b], then for any x inside of (a, b) we have
Z x
d
f (t) dt = f (x)
dx a
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If in addition g(x) is a differentiable function, by the chain rule we have
d
dx
Z
g(x)
f (t) dt = f (g(x))g 0 (x)
a
18. Find the following derivatives:
Rx 2 √
d
(2t + t) dt
(i) dx
0
R
1
d
(ii) dx
xt dt
0
R
x+x2 √
d
(iii) dx
2z + sin z dz
1
R
2
x
t2
d
(iv) dx
dt
−x2 1+t2
R
sin x 5
d
(v) dx
t dt
cos x
Rb
Rc
Rb
R4
19. Recall that for any c, we have a f (x) dx = a f (x) dx + c f (x) dx. Find 0 f (x) dx in
each of the following cases:

0≤x<1
 1
x
1≤x<2
(i) f (x) =

4−x 2≤x≤4
(ii) f (x) = |x − 2|
(iii) f (x) = 3 + |x − 3|
20. For each of the following functions f (x), find the intervals on which the graph of y = f (x)
is increasing and concave up.
Rx u
(i) f (x) = 0 √1+u
2 du
Rx
(ii) 0 cos u du
Rx
21. Let F (x) = 0 sin t dt
(i) Find F (0).
(ii) Let y = G(x). Apply the first fundamental theorem of Calculus to obtain
F 0 (x) = sin x.
(iii) Solve the differential equation
dy
dx
dy
dx
=
= sin x.
(iv) Find the solution of this equation that satisfies y(0) = F (0).
Rπ
(v) Compute F (π) = 0 sin t dt
Rπ
(vi) Use the second fundamental theorem of calculus to find 0 sin t dt and check that
you obtain the same answer as before.
Section 4.4: The second fundamental theorem of calculus and the method of substitution
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22. Compute the following definite integrals:
R4 4
(i) 1 x x−8
dx
2
R π/2
(ii) π/6 2 sin t dt
√
R0
(iii) −1 x2 x3 + 1 dx
R3 2
dx
(iv) 1 √xx3+1
+3x
R π/6
(v) 0 sin3 θ cos θ dθ
R π/6 sin θ
(vi) 0 cos
3 θ dθ
R π/2
(vii) 0 sin x sin(cos x) dx
R1
(viii) 0 x cos3 (x2 ) sin(x2 ) dx
23. Compute the following indefinite integrals:
R√
3
2x − 4 dx
(i)
R 2 3
(ii) x (x + 5) dx
√
R
2
√ x +4 dx
(iii) x sin
x2 +4
R
(iv) x2 (x3 + 5)8 cos [(x3 + 5)9 ] dx
p
R
(v) x6 sin(3x7 + 9) 3 cos(3x7 + 9) dx
p
R
(vi) x−4 sec2 (x−3 + 1) 5 tan(x−3 + 1) dx
Section 4.5: The mean value theorem for integrals and the use of symmetry
24. Find all values of c which satisfy the mean value theorem for integrals on the given
interval in each of the following cases:
√
(i) f (x) = x + 1 over [0, 3].
(ii) f (x) = sin x over [−π, π].
(iii) f (x) = |x| over [0, 2] and [−2, 2].
25. Use symmetry to help you evaluate the following integrals:
Rπ
(i) −π (sin x + cos x) dx
R π/2 sin x
(ii) −π/2 1+cos
dx
x
R π/2
(iii) −π/2 z sin2 (z 3 ) cos(z 3 ) dz
R π/4
(iv) −π/4 |x| sin5 x + |x|2 tan x dx
26. Use the substitution theorem to prove the following statements:
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(i) If f (x) is an even function (namely f (−x) = f (x) for every x), then for every a
Z a
Z a
f (x) dx = 2
f (x) dx
−a
0
(ii) If f (x) is an odd function (namely f (−x) = −f (x) for every x), then for every a
Z a
f (x) dx = 0
−a
(iii) If f (x) is a periodic function with period p (namely f (x + p) = f (x) for every x),
then
Z b+p
Z b
f (x) dx =
f (x) dx
a+p
a
Section 4.1: The area of a plane region
27. In each of the following questions, sketch the region bounded by the graphs of the given
functions and calculate the area of the region.
(i) y 2 = 4x, 4x − 3y = 4.
√
(ii) y = x, y = −x + 6, y = 0.
√
(iii) y = x, y = −x + 6, x = 0.
(iv) x = 3 − y 2 , y = x − 1.
(v) x = 3 − y 2 , y = x − 1, x = 0.
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