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Math Problems
Calculus
Find equations of tangent lines using limits
Let
f
f
f
be a function such that
f
(
−
1
)
=
3
f(-1) = 3
f
(
−
1
)
=
3
and
f
′
(
−
1
)
=
5
f'(-1) = 5
f
′
(
−
1
)
=
5
. Let
g
g
g
be the function
g
(
x
)
=
2
x
3
g(x) = 2x^3
g
(
x
)
=
2
x
3
. Let
F
F
F
be a function defined as
F
(
x
)
=
f
(
x
)
g
(
x
)
F(x) = \frac{f(x)}{g(x)}
F
(
x
)
=
g
(
x
)
f
(
x
)
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1
3
x
+
3
y
=
4
2
x
−
4
=
2
y
\begin{aligned} \dfrac{1}{3}x + 3y &= 4 \ 2x - 4 &= 2y \end{aligned}
3
1
x
+
3
y
=
4
2
x
−
4
=
2
y
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f(x)=(x+
2
2
2
)^{
2
2
2
}
−
64
-64
−
64
At what values of x does the graph of the function intersect the x-axis? Choose
1
1
1
answer:
\newline
(A)
\newline
x=
6
6
6
,x=
−
10
-10
−
10
\newline
(B)
\newline
x=
6
6
6
,x=
10
10
10
\newline
(C)
\newline
x=
−
6
-6
−
6
,x=
−
10
-10
−
10
\newline
D
\newline
f(x) does not intersect the
\newline
x-axis.
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Find the equation of the tangent line to
k
(
x
)
=
x
2
k(x) = x^2
k
(
x
)
=
x
2
at
x
=
6
x = 6
x
=
6
.
\newline
Write your answer in point-slope form using integers and fractions. Simplify any fractions.
\newline
y
−
‾
=
‾
(
x
−
‾
)
y - \underline{\quad} = \underline{\quad}(x - \underline{\quad})
y
−
=
(
x
−
)
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Find the equation of the tangent line to
k
(
x
)
=
x
2
k(x) = x^2
k
(
x
)
=
x
2
at
x
=
7
x = 7
x
=
7
.
\newline
Write your answer in point-slope form using integers and fractions. Simplify any fractions.
\newline
y
−
‾
=
‾
(
x
−
‾
)
y - \underline{\quad} = \underline{\quad}(x - \underline{\quad})
y
−
=
(
x
−
)
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Asymptotes
y
=
±
5
4
x
y = \pm \frac{5}{4}x
y
=
±
4
5
x
one vertex is
(
0
,
3
)
(0,3)
(
0
,
3
)
what is equation of hyperbola
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∠
P
∠P
∠
P
and
∠
Q
∠Q
∠
Q
are supplementary angles.
\newline
∠
P
∠P
∠
P
and
∠
R
∠R
∠
R
are vertical angles.
\newline
∠
R
∠R
∠
R
measures
3
8
∘
38^{\circ}
3
8
∘
. What is
m
∠
Q
m∠Q
m
∠
Q
in degrees?
\newline
Use the number pad to enter your answer in the box.
\newline
m
∠
Q
=
□
∘
m∠Q=\square^{\circ}
m
∠
Q
=
□
∘
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Find the area
A
A
A
of the shaded region of the cardioid
r
=
7
−
7
cos
(
θ
)
r=7-7 \cos (\theta)
r
=
7
−
7
cos
(
θ
)
.
\newline
The graph shows the graph of the cardioid with a small region shaded. The
x
\mathrm{x}
x
and
y
\mathrm{y}
y
axes are unlabelled.
\newline
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
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Function A is a linear function. An equation for Function A is
2
y
−
3
=
−
5
x
2y - 3 = -5x
2
y
−
3
=
−
5
x
. Which of the following functions has the same
y
y
y
-intercept as Function A?
\newline
Choices:
\newline
(A)
y
=
1.5
x
+
2.5
y = 1.5x + 2.5
y
=
1.5
x
+
2.5
\newline
(B)
y
=
−
1.5
x
−
2.5
y = -1.5x - 2.5
y
=
−
1.5
x
−
2.5
\newline
(C)
y
=
2.5
x
+
1.5
y = 2.5x + 1.5
y
=
2.5
x
+
1.5
\newline
(D)
y
=
−
2.5
x
−
1.5
y = -2.5x - 1.5
y
=
−
2.5
x
−
1.5
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Complete the point-slope equation of the line through
(
−
4
,
8
)
(-4,8)
(
−
4
,
8
)
and
(
4
,
4
)
(4,4)
(
4
,
4
)
.
\newline
Use exact numbers.
\newline
y
−
4
=
y-4=
y
−
4
=
□
\square
□
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Consider the curve given by the equation
\newline
x
y
2
+
5
x
y
=
50
xy^{2}+5xy=50
x
y
2
+
5
x
y
=
50
. It can be shown that
\newline
(
d
y
)
/
(
d
x
)
=
(
−
y
(
y
+
5
)
)
/
(
x
(
2
y
+
5
)
)
(dy)/(dx)=(-y(y+5))/(x(2y+5))
(
d
y
)
/
(
d
x
)
=
(
−
y
(
y
+
5
))
/
(
x
(
2
y
+
5
))
.
\newline
Write the equation of the vertical line that is tangent to the curve.
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k
2
=
m
2
+
n
2
k^2=m^2+n^2
k
2
=
m
2
+
n
2
\newline
For any right triangle, the given equation relates the length of the hypotenuse,
k
k
k
, to the lengths of the other two sides of the triangle,
m
m
m
and
n
n
n
. Which of the following equations correctly gives
m
m
m
in terms of
k
k
k
and
n
n
n
?
\newline
Choose
1
1
1
answer:
\newline
(A)
m
=
k
−
n
m=k-n
m
=
k
−
n
\newline
(B)
m
=
k
2
−
n
2
m=\sqrt{k^{2}}-n^{2}
m
=
k
2
−
n
2
\newline
(C)
m
=
k
2
−
n
2
m=\sqrt{k^{2}-n^{2}}
m
=
k
2
−
n
2
\newline
(D)
m
=
k
2
+
n
2
m=\sqrt{k^{2}+n^{2}}
m
=
k
2
+
n
2
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4
y
−
3
x
=
40
4y - 3x = 40
4
y
−
3
x
=
40
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Find the equation of the normal to the curve
y
=
x
−
2
2
x
+
1
y=\frac{x-2}{2 x+1}
y
=
2
x
+
1
x
−
2
at the point where the curve cuts the
x
x
x
-axis.
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If
y
=
(
x
−
1
)
(
x
+
5
)
y=(x-1)(x+5)
y
=
(
x
−
1
)
(
x
+
5
)
is graphed in the
x
y
x y
x
y
-plane, which of the following characteristics of the graph is displayed as a constant in the equation?
\newline
Choose
1
1
1
answer:
\newline
(A)
x
x
x
-coordinate of the vertex
\newline
(B)
x
x
x
-intercept(s)
\newline
(C) Maximum
y
y
y
-value
\newline
(D)
y
y
y
-intercept
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Given an equation
6
+
4
x
−
2
x
2
=
0
6 + 4x - 2x^2 = 0
6
+
4
x
−
2
x
2
=
0
. (a) Express
6
+
4
x
−
2
x
2
6 + 4x - 2x^2
6
+
4
x
−
2
x
2
in the form
a
−
(
x
+
b
)
2
a - (x + b)^2
a
−
(
x
+
b
)
2
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A curve in the plane is defined parametrically by the equations
\newline
x
=
t
3
+
t
x=t^{3}+t
x
=
t
3
+
t
and
\newline
y
=
t
4
+
2
t
2
y=t^{4}+2t^{2}
y
=
t
4
+
2
t
2
. An equation of the line tangent to the curve at
\newline
t
=
1
t=1
t
=
1
is
\newline
(A)
y
=
2
x
\text{(A)}\ y=2x
(A)
y
=
2
x
\newline
(B)
y
=
8
x
\text{(B)}\ y=8x
(B)
y
=
8
x
\newline
(C)
y
=
2
x
−
1
\text{(C)}\ y=2x-1
(C)
y
=
2
x
−
1
\newline
(D)
y
=
4
x
−
5
\text{(D)}\ y=4x-5
(D)
y
=
4
x
−
5
\newline
(E)
y
=
8
x
+
13
\text{(E)}\ y=8x+13
(E)
y
=
8
x
+
13
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Select all of the equations below that are equivalent to:
\newline
85
=
b
c
85 = bc
85
=
b
c
\newline
Use properties of equality.
\newline
Multi-select Choices:
\newline
(A)
5
=
b
c
17
5 = \frac{bc}{17}
5
=
17
b
c
\newline
(B)
–
5
=
b
c
–
17
–5 = \frac{bc}{–17}
–5
=
–17
b
c
\newline
(C)
–
18
=
b
c
–
5
–18 = \frac{bc}{–5}
–18
=
–5
b
c
\newline
(D)
17
=
b
c
5
17 = \frac{bc}{5}
17
=
5
b
c
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Write the formula of trigonometric ratio of
cos
(
27
0
∘
−
A
)
\cos \left(270^{\circ}-\mathrm{A}\right)
cos
(
27
0
∘
−
A
)
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Find all critical points of the function
\newline
g
(
θ
)
=
sin
2
(
6
θ
)
g(\theta)=\sin^{2}(6\theta)
g
(
θ
)
=
sin
2
(
6
θ
)
Express your answer in terms of
π
\pi
π
.
\newline
(Use symbolic notation and fractions where needed. Use
n
n
n
for all integer values.)
\newline
θ
=
\theta=
θ
=
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What is the equation of the line graphed in the
x
y
xy
x
y
-plane that passes through the point
(
−
4
,
−
5
)
(-4,-5)
(
−
4
,
−
5
)
and is parallel to the line whose equation is
3
x
−
4
y
=
−
8
3x-4y=-8
3
x
−
4
y
=
−
8
?
\newline
Choose
1
1
1
answer:
\newline
(A)
y
=
−
4
3
x
+
10
y=-\frac{4}{3}x+10
y
=
−
3
4
x
+
10
\newline
(B)
y
=
3
4
x
−
2
y=\frac{3}{4}x-2
y
=
4
3
x
−
2
\newline
(C)
y
=
3
4
x
−
8
y=\frac{3}{4}x-8
y
=
4
3
x
−
8
\newline
(D)
y
=
−
4
3
x
−
8
y=-\frac{4}{3}x-8
y
=
−
3
4
x
−
8
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Let
m
=
2
x
+
3
m=2 x+3
m
=
2
x
+
3
.
\newline
Which equation is equivalent to
(
2
x
+
3
)
2
−
14
x
−
21
=
−
6
(2 x+3)^{2}-14 x-21=-6
(
2
x
+
3
)
2
−
14
x
−
21
=
−
6
in terms of
m
m
m
?
\newline
Choose
1
1
1
answer:
\newline
(A)
m
2
−
7
m
+
6
=
0
m^{2}-7 m+6=0
m
2
−
7
m
+
6
=
0
\newline
(B)
m
2
+
7
m
+
6
=
0
m^{2}+7 m+6=0
m
2
+
7
m
+
6
=
0
\newline
(C)
m
2
+
7
m
−
15
=
0
m^{2}+7 m-15=0
m
2
+
7
m
−
15
=
0
\newline
(D)
m
2
−
7
m
−
15
=
0
m^{2}-7 m-15=0
m
2
−
7
m
−
15
=
0
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4
x
−
4
y
=
−
2
−
9
x
−
4
y
=
−
3
\begin{aligned} 4x-4y &= -2 \\ -9x-4y &= -3 \end{aligned}
4
x
−
4
y
−
9
x
−
4
y
=
−
2
=
−
3
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y
(
1
)
=
1
x
′
(
t
)
=
3
y
′
(
t
)
=
2
t
y(1)=1\quad x'(t)=3\quad y'(t)=2t
y
(
1
)
=
1
x
′
(
t
)
=
3
y
′
(
t
)
=
2
t
\newline
Write the equation of the line tangent to the curve given by
\newline
x
(
t
)
=
3
t
−
1
x(t)=3t-1
x
(
t
)
=
3
t
−
1
and
\newline
y
(
t
)
=
t
2
y(t)=t^{2}
y
(
t
)
=
t
2
at the point where
\newline
t
=
1
t=1
t
=
1
.
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Use implicit differentiation to find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
at the given point:
\newline
−
2
y
2
+
4
=
4
x
-2y^{2}+4=4x
−
2
y
2
+
4
=
4
x
at
(
−
1
,
2
)
(-1,2)
(
−
1
,
2
)
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Calculate the derivative of
y
y
y
with respect to
x
x
x
. Express derivative in terms of
x
x
x
and
y
y
y
.
\newline
e
2
x
y
=
sin
(
y
7
)
e^{2xy} = \sin(y^{7})
e
2
x
y
=
sin
(
y
7
)
\newline
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
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For the function
f
(
x
)
=
x
2
−
10
f(x)=x^{2}-10
f
(
x
)
=
x
2
−
10
, find the slope of the secant line between
x
=
−
1
x=-1
x
=
−
1
and
x
=
2
x=2
x
=
2
.
\newline
Answer:
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For the function
f
(
x
)
=
x
2
−
10
f(x)=x^{2}-10
f
(
x
)
=
x
2
−
10
, find the slope of the secant line between
x
=
−
6
x=-6
x
=
−
6
and
x
=
1
x=1
x
=
1
.
\newline
Answer:
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If
y
=
30
(
9
10
)
x
y=30\left(\frac{9}{10}\right)^x
y
=
30
(
10
9
)
x
is graphed in the
x
y
xy
x
y
-plane, which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?
\newline
Choose
1
1
1
answer:
\newline
(A) Slope
\newline
(B) The value
y
y
y
approaches as
x
x
x
decreases
\newline
(C)
x
x
x
-intercept
\newline
(D)
y
y
y
-intercept
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The function
f
f
f
is defined by
f
(
x
)
=
x
3
−
5
+
cos
(
2
x
)
f(x)=x^{3}-5+\cos (2 x)
f
(
x
)
=
x
3
−
5
+
cos
(
2
x
)
. 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
+
1
+
2
sin
(
2
x
)
f(x)=x^{3}+1+2 \sin (2 x)
f
(
x
)
=
x
3
+
1
+
2
sin
(
2
x
)
. 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
3
−
2
cos
(
2
x
2
+
2
)
f(x)=x^{3}-2 \cos \left(2 x^{2}+2\right)
f
(
x
)
=
x
3
−
2
cos
(
2
x
2
+
2
)
. 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
3
+
3
−
5
sin
(
x
2
)
f(x)=x^{3}+3-5 \sin \left(x^{2}\right)
f
(
x
)
=
x
3
+
3
−
5
sin
(
x
2
)
. 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
(
x
2
+
2
x
)
f(x)=x^{3}-2 \sin \left(x^{2}+2 x\right)
f
(
x
)
=
x
3
−
2
sin
(
x
2
+
2
x
)
. 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
+
4
x
+
3
cos
(
2
x
2
)
f(x)=x^{2}+4 x+3 \cos \left(2 x^{2}\right)
f
(
x
)
=
x
2
+
4
x
+
3
cos
(
2
x
2
)
. 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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
2
+
3
x
+
4
cos
(
3
x
)
f(x)=x^{2}+3 x+4 \cos (3 x)
f
(
x
)
=
x
2
+
3
x
+
4
cos
(
3
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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
3
+
2
x
+
3
sin
(
x
2
+
2
)
f(x)=x^{3}+2 x+3 \sin \left(x^{2}+2\right)
f
(
x
)
=
x
3
+
2
x
+
3
sin
(
x
2
+
2
)
. 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
−
5
cos
(
2
x
)
f(x)=x^{2}-5 \cos (2 x)
f
(
x
)
=
x
2
−
5
cos
(
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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
2
+
cos
(
3
x
2
)
f(x)=x^{2}+\cos \left(3 x^{2}\right)
f
(
x
)
=
x
2
+
cos
(
3
x
2
)
. 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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
3
+
3
cos
(
2
x
−
4
)
f(x)=x^{3}+3 \cos (2 x-4)
f
(
x
)
=
x
3
+
3
cos
(
2
x
−
4
)
. 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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
2
−
5
−
5
cos
(
3
x
)
f(x)=x^{2}-5-5 \cos (3 x)
f
(
x
)
=
x
2
−
5
−
5
cos
(
3
x
)
. 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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
2
+
3
cos
(
x
2
+
3
x
)
f(x)=x^{2}+3 \cos \left(x^{2}+3 x\right)
f
(
x
)
=
x
2
+
3
cos
(
x
2
+
3
x
)
. Use a calculator to write the equation of the line tangent to the graph of
f
f
f
when
x
=
−
1.5
x=-1.5
x
=
−
1.5
. You should round all decimals to
3
3
3
places.
\newline
Answer:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
2
+
5
+
3
sin
(
2
x
)
f(x)=x^{2}+5+3 \sin (2 x)
f
(
x
)
=
x
2
+
5
+
3
sin
(
2
x
)
. 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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
3
+
4
−
4
sin
(
x
2
)
f(x)=x^{3}+4-4 \sin \left(x^{2}\right)
f
(
x
)
=
x
3
+
4
−
4
sin
(
x
2
)
. 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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
2
−
2
sin
(
3
x
+
1
)
f(x)=x^{2}-2 \sin (3 x+1)
f
(
x
)
=
x
2
−
2
sin
(
3
x
+
1
)
. Use a calculator to write the equation of the line tangent to the graph of
f
f
f
when
x
=
−
1.5
x=-1.5
x
=
−
1.5
. You should round all decimals to
3
3
3
places.
\newline
Answer:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
2
+
1
−
cos
(
3
x
2
−
3
)
f(x)=x^{2}+1-\cos \left(3 x^{2}-3\right)
f
(
x
)
=
x
2
+
1
−
cos
(
3
x
2
−
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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
3
+
4
cos
(
2
x
2
)
f(x)=x^{3}+4 \cos \left(2 x^{2}\right)
f
(
x
)
=
x
3
+
4
cos
(
2
x
2
)
. 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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
2
−
sin
(
x
)
f(x)=x^{2}-\sin (x)
f
(
x
)
=
x
2
−
sin
(
x
)
. 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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
3
−
2
sin
(
x
2
)
f(x)=x^{3}-2 \sin \left(x^{2}\right)
f
(
x
)
=
x
3
−
2
sin
(
x
2
)
. 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:
Get tutor help
The function
f
f
f
is defined by
f
(
x
)
=
x
2
+
sin
(
2
x
)
f(x)=x^{2}+\sin (2 x)
f
(
x
)
=
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
=
−
1
x=-1
x
=
−
1
. You should round all decimals to
3
3
3
places.
\newline
Answer:
Get tutor help
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