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Math 1310: CSM
Sample exam 2 problems
SOLUTIONS
PLEASE NOTE that the second exam will cover all material from Chapters 2 and 3 (specifically:
Sections 2.1, 2.2, 3.1, 3.2, 3.3, 3.5, 3.6, and 3.7), and the corresponding tutorials.
There will NO be any programming questions on the exam. You certainly may want to use
your calculator for some computations, though.
1. Give the best possible estimate for g 0 (1) that can be obtained from the following table. Please
explain why you think this estimate is best.
x
0.98
.99
1
1.01
1.02
I choose
g 0 (1) ⇡
g(x)
0.8647
0.9312
1.0000
1.0712
1.1449
1.0712 0.9312
= 7.
2(0.01)
I think this is the best approximation because it corresponds to a two-sided secant line
approximation with the smallest h.
2. Let
f (x) = 3x2 + x.
(a) Find the slope of the tangent line to f (x) at the point x = 1.
f 0 (x) = 6x + 1. The slope of the line tangent to the graph of f (x) at x = 1 is f 0 (1) =
6(1) + 1 = 7.
(b) Write the equation of the tangent line to f (x) at the point x = 1. The equation is
y f (1) = f 0 (1)(x 1), or
y = 7x 3.
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Math 1310: CSM
Sample exam 2 problems
SOLUTIONS
3. For each of the following functions h(x), (i) find functions f and g such that h(x) = f (g(x));
(ii) find h0 (x).
(a) h(x) =
f (x) =
1
x+2
1
, g(x) = x + 2,
x
h0 (x) = f 0 (g(x)) g 0 (x) =
p
(b) h(x) = 3 x2 3
p
f (x) = 3 x, g(x) = x2
1
1
·1=
.
(x + 2)2
(x + 2)2
3,
1
h0 (x) = f 0 (g(x)) g 0 (x) = (x2
3
3)
2/3
· 2x =
2x
.
3(x2 3)2/3
(c) h(x) = 2x+cos(x)
f (x) = 2x , g(x) = x + cos(x),
h0 (x) = f 0 (g(x)) g 0 (x) = k2 · 2x+cos(x) · (1
(d) h(x) = 5 + tan(x4 + 3x 1)
f (x) = 5 + tan(x), g(x) = x4 + 3x
sin(x)) = k2 (1
sin(x))2x+cos(x) .
1,
h0 (x) = f 0 (g(x)) g 0 (x) = sec2 (x4 + 3x
2
1) · (4x3 + 3) = (4x3 + 3) sec2 (x4 + 3x
1).
Math 1310: CSM
Sample exam 2 problems
4. Let
f (x, y) =
(a) Find fx (x, y).
fx (x, y) =
@
@x
q
p
x
y=
SOLUTIONS
q
p
x
1 p
x
2
y.
1/2
y
@ p
( x
@x
1
y) = p pp
4 x
x
y
.
(b) Find fy (x, y).
@
fy (x, y) =
@y
q
p
x
y=
1 p
x
2
y
1/2
(c) Use the full microscope equation to estimate
q
p
4.1 1.05.
@ p
( x
@y
1
y) = pp
2
x
y
.
We let f (x, y) be as above, and choose a = 4, b = 1, x = 0.1, y = 0.05. Then
q
p
4.1 1.05 = f (a + x, b + y) = f (a, b) + z
q
p
=
4 1+ z =1+ z
0.1
0.05
pp
⇡ 1 + fx (a, b) x + fy (a, b) y = 1 + p pp
4 4
4 1 2
4 1
0.1
0.05
=1+
= 0.9875.
4·2·1 2·1
(The “actual” value is 0.987343.)
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Math 1310: CSM
Sample exam 2 problems
SOLUTIONS
5. Consider the function f (x) sketched below.
0.2
0.1
!0.5
0.5
1.0
1.5
2.0
2.5
!0.1
!0.2
On the axes below, sketch the graph of f 0 (x).
y
2.0
1.5
1.0
0.5
!0.5
0.5
1.0
1.5
!0.5
4
2.0
2.5
x