Mr. Benson
BC Calculus
Chapter 2&3 Practice Problems Answers
4
3
1) If f(x) = 5x , then what is f'(8)?
1
20 3
f'(x) =
x
3
1
20 3 20
40
f'(8) =
8 =
(2) =
3
3
3
5x2 - 3x + 1 5
=
2) Find lim
4x2 + 2x + 5 4
x
3) If f(x) =
f'(x) =
3x2 + x
, then what is f'(x)?
3x2 - x
-6x2
(3x2 - x)(6x + 1) - (3x2 + x)(6x - 1) (18x3 - 3x2 - x) - (18x3 + 3x2 - x)
=
=
2
2
2
2
(3x - x)
(3x - x)
(3x2 - x)2
4) If the function f is continuous for all real numbers and if f(x) =
lim
x 4
x2 - 7x + 12 (x - 3)(x - 4)
=
=x-3
x-4
x-4
x2 - 7x + 12
when x
x-4
4, then what is f(4)?
1
dy
?
5) If x2 - 2xy + 3y2 = 8, then what is
dx
2x - 2xy' - 2y + 6yy' = 0
6yy' - 2xy' = 2y - 2x
y'(6y - 2x) = 2y - 2x
y' =
2y - 2x
y- x
=
6y - 2x 3y - x
6) If f(x) = sec x + csc x, then what is f'(x)?
f'(x) = sec x tan x - csc x cot x
7) What is the equation of the line normal to the graph of y =
y' =
1
(6x + 2)
2 3x2 + 2x
m=
3x2 + 2x at (2, 4)?
6(2) + 2
14 7
=
= = slope of tangent
8
4
2 3(2)2 + 2(2)
equation of normal at (2, 4): y - 4 = -
4
(x - 2)
7
1
slope of normal = -
4
7
8) If f(x) = cos 2 x, then f''( ) equals what?
f'(x) = 2 cos x (-sin x) = -2cos x sin x
f''(x) = -2 cos x cos x - 2(-sin x)(sin x) = -2 cos2 x + 2 sin2 x
f''( ) = -2 cos2 + 2 sin2 = -2
9) If f(x) =
5
and g(x) = 3x, then if h(x) = g(f(x)), what is h'(2)?
2
x +1
h(x) = g(f(x)) =
15
2
x +1
h'(x) =
-30x
(x2 + 1)(0) - (2x)(15)
=
2
2
(x + 1)
(x2 + 1)2
h'(2) = -
60
12
=25
5
10) The slope of the line tangent to the graph of 3x2 + 5 ln y = 12 at (2, 1) is what?
6x +
5
y' = 0
y
y' = -
6xy
5
y' = -
6(2)(1)
12
=5
5
2
11) If f(x) = x + 5 if x < 2 , for all real numbers x, which of the following must be true?
7x - 5 if x 2
I. f(x) is continuous everywhere.
II. f(x) is differentiable everywhere.
III. The minimum value for f(x) is at x = 2.
x
lim f(x) = lim f(x)
2x 2+
x
lim f'(x) = lim f'(x)
2x 2+
Since
x
9=9
4=7
continuous
not differentiable
lim f'(x) > 0 and f(0) = 5, then f(x) does not have a minimum value at x = 2
2-
12) If f(x) = 5 3x then f'(x) is what?
f'(x) = 5 3x 3 ln 5
13) If f(x) = cos3(x + 1) then what is f'( )?
f'(x) = 3 cos2(x + 1) (-sin (x + 1)) = -3 cos2 (x + 1)sin (x + 1)
f'( ) = -3 cos2 + 1)sin ( + 1)
14) If f(x) = ln(ln(1 - x)), then f'(x)
f'(x) =
1
1
1
(
)(-1) = ln(1 - x) 1 - x
(1 - x)ln (1 - x)
2
tan (
15) lim
h 0
f'(
6
6
+ h) - tan ( )
6
is what?
h
) for f(x) = tan (x)
f'(x) = sec2(x)
f'(
6
)=
4
3
16) If f(x) = 3x2 - x and g(x) = f-1 (x), then g'(10) could be what?
10 = 3x 2 - x
x = 2 or -
5
3
g'(10) =
1
1
1
=
=
or
f'(2) 6(2) - 1 11
1
f'(-
5
)
3
=
1
1
=5
11
6(- ) - 1
3
3
17) If the function f(x) is differentiable and f(x) = ax - 6x, if x 1 , then what is a equal to?
bx2 + 4, if x > 1
x
lim f(x) = lim f(x)
1x 1+
a-6=b+4
x
lim f'(x) = lim f'(x)
1x 1+
3a - 6 = 2b
a - 10 = b
3a - 6 = 2a - 20
a = -14
18) Consider the equation x2 - 4xy + 4y2 = 64.
a) Find the expression for the slope of the curve at any point (x, y).
4y - 2x
2y - x
1
2x - 4xy' - 4y + 8yy' = 0
y' =
=
=
8y - 4x 4y - 2x 2
b) Find the equation of the tangent lines to the curve when x = 2.
4 - 8y + 4y2 = 64
y2 - 2y - 15 = 0
y = -3, 5
m at (2, -3) =
m at (2, 5) =
c) Find
1
-8
=
-16 2
8
1
=
16 2
y+3=
y-5=
1
(x - 2)
2
1
(x - 2)
2
d2 y
at (0, 4).
dx2
y'' =
(4y - 2x)(2y' - 1) - (2y - x)(4y' - 2)
=
(4y - 2x)2
(4y - 2x)(2(
2y - x
2y - x
) - 1) - (2y - x)(4(
) - 2)
4y - 2x
4y - 2x
(4y - 2x)2
3
=0
19) If 7 = xy - exy, then
dy
is what?
dx
0 = xy' + y - exy(xy' + y)
xy' - xy'exy = yexy - y
y' =
yexy - y
x - xexy
dy
equals what?
20) If y = 5t2 + 4t and x = ln t then
dx
dy
dt
dy
10t + 4
=
=
= 10t2 + 4t
dx
dx
1
dt
t
21) If a particle moves in the xy-plane so that at time t > 0 its position vector is <et2 , e-t3 >, then what its velocity
vector at t = 3?
v(t) = <2tet2 , -3t2 e-t3 >
v(3) = <6e9 , -27e-27>
22) Two particles leave the origin at the same time and move according to the equations y1 = sin 2t and y2 = 4 sin t
for 0 < t < 6. For how many values of t do the particles have the same acceleration?
v1 = 2 cos 2t
a1 = -4 sin 2t
v2 = 4 cos t
a2 = -4 sin t
-4 sin 2t = -4 sin t
sin 2t = sin t
t = 0,
3
, ,
3
23) If f and g are differentiable functions and h(x) = f(x)eg(x) (g(x) is an exponent)), then what is h'(x)?
h'(x) = f(x)eg(x)g'(x) + f'(x)eg(x) = eg(x)(f(x)g'(x) + f'(x))
tan-1 (1 + h) -
24) lim
h 0
h
4
is what?
f'(1) for f(x) = tan-1(x)
f'(x) =
1
1 + x2
f'(1) =
4
1
2
The graph of a function is given. Choose the answer that represents the graph of its derivative.
25)
(D)
26) If a particle is moving according to the vector-valued function defined by f(t) = <4t2 + 8, 6t - 2>, then what is its
speed at t = 5?
v(t) = <8t, 6>
speed = v(t) =
(8t)2 + 6 2 =
64t2 + 36
v(5) =
64(25) + 36 =
1636
27) Bonus: If f(x) = (3x)3x then f'(x) is what?
y = (3x)3x
ln y = ln (3x)3x = 3x ln 3x
1
1
y' = 3x
3 + 3 ln 3x = 3 + 3 ln 3x
y
3x
5
y' = (3 + 3 ln 3x)(3x)3x