Practice Problems for Quiz VIII
1.
(a) Carefully state Rolle’s Theorem.
(b) Carefully state the Mean Value Theorem.
(c)
Define the function G on the interval [-1, 2] as follows:
13 if  1  x  0
G ( x)  
3
13  x if 0  x  2
(d) Explain why G satisfies the hypotheses of the Mean Value Theorem on the interval [1, 2]. Sketch!
(e) Determine the value of c for the function G on the interval [-1, 2] that is guaranteed
by the Mean Value Theorem.
2. Solve each of the following initial value problems:
dy
x2

, y(2)  5
dx 1  x 3
3
dy
(b)
 1  x 4 x 5  1 , y (1)  3
dx
(a)


3. Solve the initial value problem:
dy ln x

dx
x
4.
given that y = 2014 when x = 1.
Solve the differential equation:



dx
1
1
arctan t
 tan 5 t sec 2 t 


3
dt
3  4t (5t  6)
1 t2
2
5. (a) State Rolle’s Theorem.
(b)
Using Rolle’s Theorem, prove that the function
g(x) = (x – 2) ln (x + 1) + x sin(4x)
has at least one critical point between x = 0 and x = 2? Explain!
6.
(a) State the Mean Value Theorem.
(b) Show how the Mean Value Theorem applies to the function f(x) = 4 + ln x on the
interval [1, e3]. Sketch! Find explicitly the c value.
7. Solve the initial value problem:
dy ln x

dx
x
8.
given that y = 2014 when x = 1.
Solve the differential equation:



dx
1
1
arctan t
 tan 5 t sec 2 t 


dt
3  4t (5t  6) 3
1 t2
9. Verify the formula (by differentiating):
 arcsin x dx  x arcsin x 
10.
1  x2
Verify the following anti-differentiation formula:

x sin 2 x  dx  
1
1
x cos( 2 x)  sin( 2 x)  C
2
4
3
11.
Find an anti-derivative of each of the following functions. Show your work!
(a) (1  3x) 2.9
(b)
sinh x
13  cosh x
(c )
sin x 8 cos 3 x
Hint: Express this as a function of sin x multiplied by cos x.
(d )
cos x sin 3 x
Hint:
Similar to (c).
12. Find the indefinite integral of each of the following functions.
(a)
(b)
(c )
13.
1  4 ln x
x
x3e58 x
4
arctan x
1 x2
Solve the following initial value problem:
dy 3
 t cos(t 4 )  t  4
dt
14.
Show your work!
given that y = 3 when t = 0.
The figure below is called a Lissajous figure. (Lissajous curves have applications in
physics, astronomy, and other sciences.)
x(t) = sin 2t
y(t) = sin 3t
It is parameterized by the equations:
4
(a)
Find the point in the interior of the first quadrant where the tangent to the
curve is horizontal.
(b)
Find the equations of the two tangent lines at the origin.
15. Below is the graph of the derivative, F(x), of a function F(x).
(a)
Sketch the graph of F[x].
(b)
Sketch the graph of F[x]. Indicate local max/min, regions of
increase/decrease, regions where F is concave up/down, and all inflection
points.
6
4
2
1
2
2
3
4
5
6
7
5
16. Using the method of judicious guessing, find an anti-derivative of each of the following
functions:
1
 x3
a)
  x3 
b)
cosh x – 3 sin 3x + sec2 (4x)
c)
ex – e4
d)
x3  x2  x  1
x
e)
( x  3)( x  4)
f)
F(x) = a x4 + b sin cx
g)
F(x) = x sin(x2) + 1
h)
F(x) = sec2(x) + (sec x)(tan x)
i)
F(x) = 9 cosh x
j)
F(x) = sin(4x) + e9x
k)
F(x) = x4(3 + 4x5)6
l)
m)
F ( x) 
F ( x) 
ex
1  e2x
1
x4  5 ln x 
17. Can the following limit be solved using L’Hôpital’s rule? Explain.
3x 2  1
lim
x 1 2 x  3
6
Can the following limit be solved using L’Hôpital’s rule? Explain.
18.
sin x
x 0 x 2  3x
lim
Can the following limit be solved using L’Hôpital’s rule? Explain.
19.
cos 2 x  1
x 0 x3  5 x 2
lim
Can the following limit be solved using L’Hôpital’s rule? Explain.
20.
ex
lim 13
x  x
Can the following limit be solved using L’Hôpital’s rule? Explain.
21.
x
x  0 ln x
lim
22. (MIT 18.01 final)
Use L’Hôpital’s rule to compute the following limits:
(a)
(b)
ax  bx
lim
w h ere0  a  b
x 0
x
4 x3  5 x  1
lim
x 1
ln x
7
23. Using de l’Hôpital’s rule, compute the following limit:
lim
x 0
e2 x  1  2x  2x2
x3
A man is like a fraction whose numerator is what he is and whose
denominator is what he thinks of himself. The larger the
denominator, the smaller the fraction.
– Tolstoy