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Resources

Find derivatives of sine and cosine functions

Find the volume of the solid generated when the region enclosed by

y=-\sqrt{x+1}, y=0

x=2

is revolved about the

\mathrm{x}

Find the particular solution,

y=f(x)

, to the differential equation

\frac{d y}{d x}=\frac{\cos x}{y}

f\left(\frac{3 \pi}{2}\right)=-1

Evaluate the expression, given functions

f

g

\newline

\begin{array}{l} f(x)=3 x-2 \\ g(x)=5-x^{2} \end{array}

\newline

3 f(1)-4 g(-2)=

Choose the equation which describes the transformation of

y=f(x)

represented by the following graph.

\newline

\begin{array}{r} y=f(2 x)[] \\ y=f(x-1)[] \\ y=f(x)-1[] \\ y=-f(x)[] \\ y=f(x)+1[] \\ y=f(-x)[] \\ y=\frac{1}{2} f(x)[] \end{array}

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10

. Copper (I) sulfide reacts with sulfur to produce copper (II) sulfide at

25^{\circ} \mathrm{C}

. The process is exothermic

\left(\Delta \mathrm{H}^{\circ}\right.

=-26.7 \mathrm{~kJ} / \mathrm{mol})

\Delta G^{\circ}

\newline

\mathrm{Cu} 2 \mathrm{~S}(\mathrm{~s})+\mathrm{S}(\mathrm{s}) \rightarrow 2 \mathrm{CuS}(\mathrm{s})

2

()

sulfide macts with sulfur to produce copper (II) sulfide at

25^{\circ}C

. The process is exothermic

\Delta H^{\circ}:

16^{\circ}

\newline

\text{Cu}_\text{s}(\text{s})+\text{S}(\text{s})\rightarrow 2\text{Cu}_\text{s}(\text{s})

9

. Use a standard enthulpies of formation table to determine the delta If for each of these reactions. State whether each reaction is endothermic or exothermic.

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2 \mathrm{CO}(\mathrm{g})+\mathrm{O} 2(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_{2}(\mathrm{~g})

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2 \mathrm{H} 2 \mathrm{~S}(\mathrm{~g})+3 \mathrm{O} 2(\mathrm{~g}) \rightarrow 2 \mathrm{H} 2 \mathrm{O}(\mathrm{l})+2 \mathrm{SO} 2(\mathrm{~g})

\begin{array}{l}D H=12 \\ A C=15 \\ D B=13 \\ A B=?\end{array}

\begin{array}{l}D H=12 \\ A C=15 \\ P B=13 \\ A B=?\end{array}

P_{\mathrm{o}}=\left\{\begin{array}{ll}\frac{(1-m) V_{\mathrm{ds}}^{2}}{2} d^{2} & \text { DCM: } d<\frac{m}{2} \\ \frac{m V_{\mathrm{sc}}^{2}}{L_{\mathrm{cq}} f_{\mathrm{sw}}}\left(\frac{d}{2}(1-d)-\frac{m^{2}}{8}\right) & \text { CCM }: d \geq \frac{m}{2}\end{array}\right.

1

. Given a function

f: \mathbb{R}^{2} \rightarrow \mathbb{R}

, the first-order partial derivatives are provided:

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\frac{\partial f}{\partial x}(x, y)=6 x-6 y \quad \text { and } \quad \frac{\partial f}{\partial y}(x, y)=6 y^{2}-6 x .

\newline

(a) Determine all stationary points of

f

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(b) Compute the second-order partial derivatives of

f

and find the Hessian matrix

\boldsymbol{H}_{f}(x, y)

f

. Determine the points in the

(x, y)

f

has a local maximum, a local minimum or a saddle point.

\newline

(c) It is now stated that

f(0,0)=1

. Determine the second-degree Taylor polynomial

P_{2}(x, y)

f

with the expansion point

f

0

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2

. Define the function

f

1

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f(x)=\left\{\begin{array}{ll} \frac{\sin (x)}{x} & x \neq 0, \\ 1 & x=0 . \end{array}\right.

\newline

(a) Find the third-degree Taylor polynomial

f

2

f

3

with expansion point

f

4

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\newline

\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=1 .

\newline

Hint: Use (a) and Taylor's limit formula.

\newline

f

is continuous on

f

6

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(d) Compute, e.g. using SymPy, a decimal approximation of

f

7

. You should include at least

5

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(e) Compute a Riemann sum

f

8

f

7

, where we require that

f

0

f

1

\newline

\text{CO}_{2(g)}

at STP is obtained by complete combustion of

6\,\text{g}

\newline

2023

\newline

22.4\,\text{dm}^{3}

\newline

11.2\,\text{dm}^{3}

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5.6\,\text{dm}^{3}

\newline

2.24\,\text{dm}^{3}

\begin{array}{c} L A=\pi r l \\ S A=\pi r l+\pi r^{2} \end{array}

\newline

r=

\newline

l=

E(a X+b)=a E(X)+b

E(X)

is the expected value of a discrete random variable

X

a

b

\newline

\operatorname{Var}(a X+b)=a^{2} \operatorname{Var}(X)

Let's try to find a rotiano approximation for

\sqrt{ } 2

\newline

2

\newline

\begin{aligned} \sqrt{2} \times \sqrt{2} & =2 \\ n \times n & \approx 2 \end{aligned}

[SA~~],[V=],[HCS=]

Given the inscribed polygon, find the value of both

x

y

.\begin{align*}x&=\y&=\end{align*}

\begin{array}{r}f(x)=8 e^{x \cos x} \\ f^{\prime}(x)=\end{array}

Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented counterclockwise

\newline

The circulation line integral of

F=\left\langle 3 x y^{2}, 3 x^{3}+y\right\rangle

C

is the boundary of

\{(x, y): 0 \leq y \leq \sin x, 0 \leq x \leq \pi\}

Extraneous solutions of equations

\newline

Which value for the constant

c

z=2

an extraneous solution in the following equation?

\newline

\begin{array}{l} \sqrt{7-3 z}=3+c z \\ c=\emptyset \end{array}

f(x)

g(x)

are differentiable. The function

h(x)

h(x)= \frac{f(x)}{g(x)}

f(-1)= 6

f'(-1)= -3

g(-1)= 2

g'(-1)= 5

h'(-1)

Find the following values of the function

\newline

\begin{array}{l} f(x)=\left\{\begin{array}{ll} x+2 & x \leq 5 \\ 6-x & x>5 \end{array}\right. \\ f(2)=\square \\ f(5)=\square \\ f(11)=\square \end{array}

Determine whether or not

F

is a conservative vector field. If It is, find a function

f

such that F = \(\newlineabla f\). (If the vector field is not conservative, enter DNE.)

\newline

\(\mathbf{F}(x,y)=\langle y e^{x}+\sin(y), e^{x}+x \cos(y) \rangle

Determine whether or not

F

is a conservative vector field. If it is, find a function

f

such that F=\(\newlineabla f\). (If the vector field is not conservative, enter DNE.

\newline

\begin{cases} F(x,y)=e^{x}\sin(y)i+e^{x}\cos(y)j \end{cases}

f(x,y)=

Determine whether or not

\mathbf{F}

is a conservative vector field. If it is, find a function

f

such that F=\(\newlineabla f \). (If the vector field is not conservative, enter DNE.)

\newline

F(x, y)=\square)=\left(y e^{x}+\sin (y)\right) \mathbf{i}+\left(e^{x}+x \cos (y)\right) \mathbf{j}

Evaluate the line integral, where

C

is the given space curve.

\newline

\int_{C} z^{2} d x+x^{2} d y+y^{2} d z, C

is the line segment from

(1,0,0)

(3,1,4)

\cos x=\frac{3}{5}, x

I

, then find (without finding

x

\newline

\begin{array}{l} \sin (2 x)= \\ \cos (2 x)= \\ \tan (2 x)= \end{array}

\sin x=\frac{2}{3}, x

I

, then find the exact answers for the following (without finding

x

\newline

\begin{array}{l} \sin (2 x)= \\ \cos (2 x)= \\ \tan (2 x)= \end{array}

Which of the following equations represent functions where

x

is the input and

y

is the output? Select all that apply.

\newline

Multi-select Choices:

\newline

y = 1 - x

\newline

y = x

\newline

x - y = 1

\newline

x = 1

\newline

y = -x - 1

Which of the following equations represent functions where

x

is the input and

y

is the output? Select all that apply.

\newline

Multi-select Choices:

\newline

y = 0.25x

\newline

y = \frac{2}{5}

\newline

x - y = 25

\newline

y = 25 - 5x

\newline

y = 5^x

9

- Avg value and arc len (

1

\newline

Find the average value of the functions on the given interval.

\newline

a) Average value of

f(x)=x

[5,13]

Evaluate the line integral, where

C

is the given curve.

\newline

\int_{C} y^{3} d s, \quad C: x=t^{3}, \quad y=t, \quad 0 \leq t \leq 2

\newline

m \angle A B D

is a straight angle.

\newline

\begin{array}{l} m \angle A B C=8 x-41^{\circ} \\ m \angle C B D=9 x+17^{\circ} \end{array}

\newline

m \angle C B D

\left\{\begin{array}{l}y z-\lambda_{1}-\lambda_{2}=0 \\ x z-\lambda_{1}+\lambda_{2}=0 \\ x y-\lambda_{2}=0 \\ -(x+y-2)=0\end{array}\left\{\begin{array}{l}\lambda_{2}=-\lambda_{1}+y z=\lambda_{1}-x_{2} \\ \lambda_{2}=\lambda_{1}(-z+y)+y \\ \lambda_{2}=-x y\end{array}\left\{\lambda_{1}\right.\right.\right.

\begin{array}{l}y=-x^{2}-6 x-3 \\ x=-\frac{b}{2 a}\end{array}

g

be the function given by

g(x)=\int_{-1}^{x} f(t) d t

g(1

\newline

f o s=\left\{\begin{array}{lll} 3 x^{2}+2 x & \text { for } & x \leq 0 \\ e^{2 x}+2 & \text { for } & x>0 \end{array}\right.

g

be the function given by

g(x)=\int_{-1}^{x} f(t) d t

g(1)

\newline

f_{0} s=\left\{\begin{array}{lll} 3 x^{2}+2 x & \text { for } & x \leq 0 \\ e^{2 x}+2 \text { for } & x>0 \end{array}\right.

x^{4} \leq f(x) \leq x^{2}

x

[-1,1]

x^{2} \leq f(x) \leq x^{4}

x < -1

x > 1

, at what points

c

do you automatically know

\lim_{x \rightarrow c}f(x)

? What can you say about the value of the limit at these points?

Evaluate the expression.

\newline

\begin{array}{c} a=8 \quad b=4 \\ a b+2 b-3 a \end{array}

Which balanced equation represents a redox reaction?

\newline

1

\mathrm{AgNO}_{3(\mathrm{aq})}+\mathrm{NaCl}_{(\mathrm{aq})} \rightarrow \mathrm{AgCl}_{(\mathrm{s})}+\mathrm{NaNO}_{3(\mathrm{aq})}

\newline

2

\mathrm{H}_{2} \mathrm{CO}_{3(\mathrm{aq})} \rightarrow \mathrm{H}_{2} \mathrm{O}_{(\ell)}+\mathrm{CO}_{2(\mathrm{~g})}

\newline

3

\mathrm{NaOH}_{(\mathrm{aq})}+\mathrm{HCl}_{(\mathrm{aq})} \rightarrow \mathrm{NaCl}_{(\mathrm{aq})}+\mathrm{H}_{2} \mathrm{O}_{(\ell)}

\newline

4

\mathrm{Mg}_{\text {(s) }}+2 \mathrm{HCl}_{\text {(aq) }} \rightarrow \mathrm{MgCl}_{\text {(aq) }}+\mathrm{H}_{2(\mathrm{~g})}

\newline

1

\newline

2

\newline

3

\newline

4

For the function

f(x)

given below, evaluate

\newline

\begin{array}{l} \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x) . \\ \quad f(x)=5 \sin \left(4 x^{3}\right) \end{array}

For the function

f(x)

given below, evaluate

\lim _{x \rightarrow \infty} f(x)

\lim _{x \rightarrow-\infty} f(x)

\newline

f(x)=5 \sin \left(4 x^{3}\right)

17

. In the circuit shown above, the emf's of the batteries are given, as well as the currents in the outside and the resistance in the middle branch. What is the magnitude of the potential difference between

X

\newline

4 \mathrm{~V}

\newline

8 \mathrm{~V}

\newline

10 \mathrm{~V}

\newline

12 \mathrm{~V}

\newline

16 \mathrm{~V}

9

\newline

\newline

In the plane referred to an orthopormal system, given the curve

\left(C^{\prime}\right)

representing the derivative function

f^{\prime}

f

differentiable over

\mathbb{R}

\newline

1

) Using the curve

(C)

, determine with justification:

\newline

a) The sense of variations of the function

f

\mathbb{R}

\newline

b) The convexity of the function

f

\mathbb{R}

\newline

2

) Suppose that the function

f

is defined over

\mathbb{R}

f^{\prime}

1

f^{\prime}

2

is a real number.

\newline

f^{\prime}

3

, the derivative function of

f

\mathbb{R}

as a function of

f^{\prime}

2

\newline

b) Determine graphically

f^{\prime}

7

and then deduce the value of

f^{\prime}

2

Solve the equation.

\newline

\begin{array}{l} \frac{1}{2}=\frac{2}{3} x \\ x=\square \end{array}

\begin{array}{ll}x^{3} & -x(y-z)^{2} \\ y^{3} & -y(z-x)^{9} \\ z^{3} & -z(x-y)^{9}\end{array}

\begin{array}{l}\frac{3 y}{2}-7=2 \\ y=?\end{array}

jolve the system of equ

\newline

\begin{array}{c} 4 x-3 y=-4 \\ 4 x+y=12 \end{array}

\begin{array}{l}\tan x=\frac{7}{24} \\ \sec x=\frac{-25}{24}\end{array}

2

p t s

.) Find the area of the shaded region, given that

m \angle A C B=90^{\circ}, A O=10 \mathrm{~cm}

A C=12 \mathrm{~cm}

\newline

\begin{array}{c} A B=20 \mathrm{~cm} \\ A C=12 \mathrm{~cm} \\ C B= \\ a^{2}+b^{2}=c^{2} \end{array}

...