MIDTERM 2
Form A, Page 2
SOLUTIONS
1. (16 pts)
(a) (6 pts) Fill in the blanks.
f' (x) = lim
h→ 0
f (x + h) - f (x)
h
if the limit exists.
(b) (10 pts)
Let f (x) =
1
.
x+4
Use the (limit) definition of derivative in (a) to find
f' (x).
DO NOT USE THE PRODUCT OR QUOTIENT RULE ! SHOW YOUR WORK !
f' (x) = lim
h→ 0
= lim
h→ 0
f (x + h) - f (x)
= lim
h→ 0
h
1
1
x+h+4
x+4 =
h
x + 4 - (x + h + 4)
- h
= lim
=
h→
0
h (x + h + 4) (x + 4)
h (x + h + 4) (x + 4)
= lim
h→ 0
- 1
- 1
- 1
=
=
(x + h + 4) (x + 4) (x + 4) (x + 4) (x + 4)2
2
SOLUTIONS-MIDTERM 2-AU15.nb
MIDTERM 2
Form A, Page 3
2. (18 pts) A part of the curve with equation
cos (πxy) +x +y = 1
is sketched below.
(a) Use implicit differentiation to find the derivative
d
dx
 cos (πxy) + x + y = 1] ⟹ -sin (πxy)
-sin (πxy)πy + πx
dy
dx
 +1 +
- πy sin (πxy) -πx sin (πxy)
dy
dx
dy
dx
1 - πx
=
dy
dx
dy
dx
d
dy
dx
.
(πxy) +1 +
dx
dy
dx
=0
+1 +
dy
dx
=0
sin (πxy)] = πy sin (πxy) - 1
πy sin (πxy) - 1
1 - πx sin (πxy)
(b) Consider the point (1, 1). Show (algebraically) that this
point lies on the curve.
cos (π (1) (1)) + 1 + 1 = cos (π) + 2 = -1 + 2 = 1
✓
Find the equation of the line tangent to the curve at
this line in the figure above.
=0
SOLUTIONS-MIDTERM 2-AU15.nb
(c) Find the equation of the line tangent to the curve at (1,
Draw this line in the figure above.
1 ).
y - 1 = m (x - 1) where
m=
dy
π (1) sin (π (1) (1)) - 1

=
dx x=1
1 - π (1) sin (π (1) (1))
=
y=1
π sin (π) - 1
1 - π sin (π)
=
0 - 1
= -1
1 -0
So, the equation of the line tangent to the curve at (1,
y - 1 = -(x - 1)
1) is
3
4
SOLUTIONS-MIDTERM 2-AU15.nb
3. (18 pts)
MULTIPLE CHOICE!!!
A table of values for f (x) and f' (x) is shown below.
Suppose that f is a one - to - one function and
f-1 (x) is its inverse.
x f (x) f ' (x)
1
3
4
3
4
6
4
5
3
CIRCLE THE CORRECT ANSWER IN EACH PART.
(I) Evaluate f-1 (f (x)) at x = 3.
f-1 (f (x)) = x, for all x in the domain of f.
(a) 1 ;
(e) DOES NOT EXIST ;
d
(II) Evaluate
d
dx
d
(d) 4 ;
f NONE OF THE PREVIOUS ANSWERS.
f (f (x)) at x = 3.
dx
f (f (x)) = f' (f (x)) f' (x);
dx
f (f (x)) = f' (f (3)) f' (3) = f' (4) (5) = (3) (5) = 15
x=3
(a) 6 ;
(b) 25 ;
(e) DOES NOT EXIST ;
(III) Evaluate
d
dx
(a)
(c) 6 ;
(b) 3 ;
1
4
;
d
dx
(d) 15
;
f NONE OF THE PREVIOUS ANSWERS.
ln (f (x)) at x = 3.
ln (f (x)) =
x=3
(b) 5 ;
(e) DOES NOT EXIST ;
(c) 5 ;
1
5
f ' (3) =
f (3)
4
(c)
5
;
4
(d)
1
5
f NONE OF THE PREVIOUS ANSWERS.
;
SOLUTIONS-MIDTERM 2-AU15.nb
3. (CONTINUED)
(IV)
Evaluate f-1 (x) at x = 3.
f-1 (3)
= 1⟺
(a) 4 ;
f (1) = 3
(b) 1 ;
(e) DOES NOT EXIST ;
(V) Evaluate
d
dx x=3
d
dx
5;
1
3
(d) 5;
;
f NONE OF THE PREVIOUS ANSWERS.
f-1 (x) at x = 3.
f-1 (x) =
(a) 1 ;
(e)
(c)
1
f ' (f-1 (3)
(b) 4 ;
=
(c)
1
1
=
f ' (1)
4
1
5
;
(d)
1
;
4
f NONE OF THE PREVIOUS ANSWERS.
(VI) Find the average rate of chnage of f over the interval [1, 3].
Average rate of chnage of f over the interval [1, 3]
f (3) - f (1)
4- 3
1
=
=
=
3-1
3-1
2
(a)
2;
(e) DOES NOT EXIST ;
(b) 1 ;
(c)
1
2
;
(d) 5 ;
f NONE OF THE PREVIOUS ANSWERS.
5
6
SOLUTIONS-MIDTERM 2-AU15.nb
4. (30 pts) EXPLANATION IS NOT REQUIRED, AND NO PARTIAL CREDIT WILL BE GIVEN.
(I) The (entire) graph of a function f is shown in the figure below.
f(x)
2
1
1
-1
2
3
4
5
x
-1
-2
(a) Find the x - coordinates of all critical points of f (or write NONE).
ANSWER : critical point (s) at
x = 0, 1, 3
(b) Find the x - coordinates of all local minima of f (or write NONE).
ANSWER : local min (s) at x = 3
(c) Find all values of x at which f attains its global minimum (or write NONE).
ANSWER : global min (s) at x = NONE
(d) Find the interval (or intervals) on which the derivative of f is increasing.
ANSWER : derivative of f is increasing on (0, 1)
(II) The figure below shows the graphs of f, f', and f ".
y
A
2
1
B
Which curve is which?
-2
1
-1
2
x
C
-1
Fill in the blanks: A =
f;
B=
f ' ;,
C=
f ''
.
SOLUTIONS-MIDTERM 2-AU15.nb
4. EXPLANATION IS NOT REQUIRED, AND NO PARTIAL CREDIT WILL BE GIVEN.
f ' derivative of f
f' (x) = (x - 2) ex .
(III) A function
is given
The following questions are about the function f.
(a) Find all critical points of f (or write NONE).
ANSWER : critical point (s) at x = 2
(b) On what interval (or intervals) is f increasing?
ANSWER : f is increasing on
(2, + ∞ )
(c) Find the point or points where f has a local maximum (or write NONE).
ANSWER : local max at x =
NONE
(d) Find the point or points where f has a local minimum.
ANSWER : local min (s) at x =
2
(e) Find f "(x), the second derivative of f.
ANSWER : f'' (x) =
ex + (x - 2) ex = (x - 1) ex
(f) Identify any inflection pionts.
ANSWER : inflection point (s) at x = 1
(g) Determine the intervals on which the function is concave up
or concave down.
ANSWER : f is CONCAVE UP on (1, +∞)
f is CONCAVE DOWN on (-∞, 1)
7
8
SOLUTIONS-MIDTERM 2-AU15.nb
MIDTERM 2
Form A, Page 8
5. (18 pts) A right cone has fixed slant height (see figure) of 9 ft.
The cone' s height is shrinking at a rate of 0.5 ft / sec.
At what rate is the area of the base changing when the height is 6 ft?
Make sure to label the picture.
A =
r2
r2
π
+ h2 = 92 ⟹ r2 = 92 - h2 ⟹ A = r2 π = ( 92 - h 2 ) π
So,
A = ( 92 - h2 ) π
dA
dh
= -2 π h
dt
dt

and
dA

= -2 π (6 ft) (-0.5 ft / sec).
dt h=6
Therefore,

and
dA
ft2

=6π
dt h=6
sec