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Math Problems
Calculus
Find equations of tangent lines using limits
The function
f
f
f
is defined by
f
(
x
)
=
x
3
+
cos
(
2
x
2
+
5
)
f(x)=x^{3}+\cos \left(2 x^{2}+5\right)
f
(
x
)
=
x
3
+
cos
(
2
x
2
+
5
)
. Use a calculator to write the equation of the line tangent to the graph of
f
f
f
when
x
=
1
x=1
x
=
1
. You should round all decimals to
3
3
3
places.
\newline
Answer:
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The function
f
f
f
is defined by
f
(
x
)
=
x
3
+
2
sin
(
3
x
−
3
)
f(x)=x^{3}+2 \sin (3 x-3)
f
(
x
)
=
x
3
+
2
sin
(
3
x
−
3
)
. Use a calculator to write the equation of the line tangent to the graph of
f
f
f
when
x
=
0.5
x=0.5
x
=
0.5
. You should round all decimals to
3
3
3
places.
\newline
Answer:
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The function
f
f
f
is defined by
f
(
x
)
=
x
2
−
2
x
+
3
cos
(
x
2
−
x
)
f(x)=x^{2}-2 x+3 \cos \left(x^{2}-x\right)
f
(
x
)
=
x
2
−
2
x
+
3
cos
(
x
2
−
x
)
. Use a calculator to write the equation of the line tangent to the graph of
f
f
f
when
x
=
−
2.5
x=-2.5
x
=
−
2.5
. You should round all decimals to
3
3
3
places.
\newline
Answer:
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The function
f
f
f
is defined by
f
(
x
)
=
x
2
+
x
−
2
sin
(
2
x
)
f(x)=x^{2}+x-2 \sin (2 x)
f
(
x
)
=
x
2
+
x
−
2
sin
(
2
x
)
. Use a calculator to write the equation of the line tangent to the graph of
f
f
f
when
x
=
3
x=3
x
=
3
. You should round all decimals to
3
3
3
places.
\newline
Answer:
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The exponential function
f
f
f
is graphed in the
x
y
x y
x
y
-plane. As
x
x
x
increases by
1
,
y
1, y
1
,
y
increases by a factor of
3
3
3
. Which of the following could be
f
f
f
?
\newline
Choose
1
1
1
answer:
\newline
(A)
f
(
x
)
=
(
1
3
)
x
f(x)=\left(\frac{1}{3}\right)^{x}
f
(
x
)
=
(
3
1
)
x
\newline
(B)
f
(
x
)
=
(
1
3
)
x
+
3
f(x)=\left(\frac{1}{3}\right)^{x}+3
f
(
x
)
=
(
3
1
)
x
+
3
\newline
(C)
f
(
x
)
=
3
x
+
2
f(x)=3^{x}+2
f
(
x
)
=
3
x
+
2
\newline
(D)
f
(
x
)
=
2
(
3
)
x
f(x)=2(3)^{x}
f
(
x
)
=
2
(
3
)
x
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The learning rate for new skills is proportional to the difference between the maximum potential for learning that skill,
M
M
M
, and the amount of the skill already learned,
L
L
L
.
\newline
Which equation describes this relationship?
\newline
Choose
1
1
1
answer:
\newline
(A)
L
(
t
)
=
k
(
M
−
L
)
L(t)=\frac{k}{(M-L)}
L
(
t
)
=
(
M
−
L
)
k
\newline
(B)
L
(
t
)
=
k
(
M
−
L
)
L(t)=k(M-L)
L
(
t
)
=
k
(
M
−
L
)
\newline
(C)
d
L
d
t
=
k
(
M
−
L
)
\frac{d L}{d t}=k(M-L)
d
t
d
L
=
k
(
M
−
L
)
\newline
(D)
d
L
d
t
=
k
(
M
−
L
)
\frac{d L}{d t}=\frac{k}{(M-L)}
d
t
d
L
=
(
M
−
L
)
k
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Consider the curve given by the equation
3
x
2
+
y
4
+
6
x
=
253
3 x^{2}+y^{4}+6 x=253
3
x
2
+
y
4
+
6
x
=
253
. It can be shown that
d
y
d
x
=
−
6
(
x
+
1
)
4
y
3
.
\frac{d y}{d x}=\frac{-6(x+1)}{4 y^{3}}.
d
x
d
y
=
4
y
3
−
6
(
x
+
1
)
.
\newline
Write the equation of the horizontal line that is tangent to the curve and is above the
x
x
x
-axis.
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Consider the curve given by the equation
x
y
2
+
5
x
y
=
50
x y^{2}+5 x y=50
x
y
2
+
5
x
y
=
50
. It can be shown that
d
y
d
x
=
−
y
(
y
+
5
)
x
(
2
y
+
5
)
.
\frac{d y}{d x}=\frac{-y(y+5)}{x(2 y+5)} \text {. }
d
x
d
y
=
x
(
2
y
+
5
)
−
y
(
y
+
5
)
.
\newline
Write the equation of the vertical line that is tangent to the curve.
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What is the area of the region between the graphs of
f
(
x
)
=
x
+
1
f(x)=\sqrt{x+1}
f
(
x
)
=
x
+
1
and
g
(
x
)
=
2
x
−
4
g(x)=2 x-4
g
(
x
)
=
2
x
−
4
from
x
=
0
x=0
x
=
0
to
x
=
3
x=3
x
=
3
?
\newline
Choose
1
1
1
answer:
\newline
(A)
23
3
\frac{23}{3}
3
23
\newline
(B)
5
3
\frac{5}{3}
3
5
\newline
(C)
14
3
\frac{14}{3}
3
14
\newline
(D)
−
3
-3
−
3
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What is the area of the region between the graphs of
f
(
x
)
=
x
2
+
12
x
f(x)=x^{2}+12 x
f
(
x
)
=
x
2
+
12
x
and
g
(
x
)
=
3
x
2
+
10
g(x)=3 x^{2}+10
g
(
x
)
=
3
x
2
+
10
from
x
=
1
x=1
x
=
1
to
x
=
4
x=4
x
=
4
?
\newline
Choose
1
1
1
answer:
\newline
(A)
77
77
77
\newline
(B)
64
3
\frac{64}{3}
3
64
\newline
(C)
18
18
18
\newline
(D)
45
45
45
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The graphs of the functions
f
(
x
)
=
sin
(
x
)
f(x)=\sin (x)
f
(
x
)
=
sin
(
x
)
and
g
(
x
)
=
1
2
g(x)=\frac{1}{2}
g
(
x
)
=
2
1
intersect at
2
2
2
points on the interval
0
<
x
<
π
0<x<\pi
0
<
x
<
π
.
\newline
What is the area of the region bound by the graphs of
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
between those points of intersection ?
\newline
Choose
1
1
1
answer:
\newline
(A)
π
3
\frac{\pi}{3}
3
π
\newline
(B)
π
2
\frac{\pi}{2}
2
π
\newline
(C)
2
−
π
2
2-\frac{\pi}{2}
2
−
2
π
\newline
(D)
3
−
π
3
\sqrt{3}-\frac{\pi}{3}
3
−
3
π
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