A vehicle moves along a straight road. The vehicle's position is given by $f(t)$, where $t$ is measured in seconds since the vehicle starts moving. During the first 10 seconds of the motion, the vehicle's acceleration is proportional to the cube root of the time since the start. Which of the following differential equations describes this relationship, where $k$ is a positive constant? A $\frac{d f}{d t}=k \sqrt[3]{t}$ (B) $\frac{d f}{d t}=k \sqrt[3]{f}$ C $\frac{d^{2} f}{d t^{2}}=k \sqrt[3]{t}$ (D) $\frac{d^{2} f}{d t^{2}}=k \sqrt[3]{f}$

The radius of a spherical tumor is growing by $\frac{1}{4}$ centimeter per week. Find how rapidly the volume is increasing at the moment when the radius is 5 centimeters. [Hint: The volume of a sphere of radius $r$ is $V=\frac{4}{3} \pi r^{3}$.] (Round your answer to the nearest integer.) \[ \mathrm{cm}^{3} \text { per week } \]

Identify the inside function, $u=g(x)$, and the outside function, $y=f(u)$. \[ \begin{aligned} y & =\left(x^{2}-7 x+9\right)^{4} \\ u=g(x) & =2 x-7 \\ y & =f(u)= \end{aligned} \]Identify the inside function, u = g(x), and the outside function, y = f(u). y = (x^2 − 7x + 9)^4

1. For the following functions, find all critical points, and classify whether they are local maximum, local minimum, or saddle points. (a) $f(x, y)=x^{2} y+\frac{4}{3} y^{3}-4 y$. (b) $f(x, y)=x^{3}+y^{3}+3 x^{2}-3 y^{2}-8$. (c) $f(x, y, z)=x^{3}+x y z+y^{2}-3 x$.

An appliance company wants to investigate the life expectancy and reliability of a certain dishwasher. We use a probability density function f(x), where f(x) ≥ 0 for all x and where ∫ b a f(x) dx represents the probability that the dishwasher breaks down during the time interval from x = a to x = b, where x is measured in years. For the life expectancy of a dishwasher, we use the interval [0, ∞) because we do not consider negative time.   (a) (3 marks) A probability density function on the interval [0, ∞) needs to have the property that ∫ ∞ 0 f(x) dx = 1, so that the total probability is 1 (meaning 100%). Suppose the tech company estimates that the life expectancy of the dishwasher has probability density function fd(x) = ke−0.07x for some value of the constant k. Find the value of k that makes fd(x) = ke−0.07x a probability density function.   (b) (3 marks) One measure of ‘average’ is the mean. The mean (denoted µ) of a random variable with probability density function f(x) on [0, ∞) is defined by µ = ∫ ∞ 0 xf(x) dx. Find the mean life expectancy of the dishwasher (using your fd(x) from part (a)). Convert your answer to number of years and months (round to the nearest month).   (c) (2 marks) Find the probability of the dishwasher lasting 10 years or longer round to 2 decimal places, or equivalently an integer percent. Optional (not part of the assignment): Compare your answers to parts (b) and (c). Do your answers surprise you? Note that there are several different ways to measure ‘average’!   (d) (2 marks) The Reliability function R(t) of the dishwasher is defined as the probability that the dishwasher lasts at least t years. Find the Reliability function for your fd(x) from part (a).

A specimen of charcoal found at an archaeological site is estimated to be approximately 4480 years old according to radiocarbon dating. The half-life period for radioactive decay of ${ }^{14} \mathrm{C}$ is approximately 5700 years. What percentage of the ${ }^{14} \mathrm{C}$ atoms must have been found in the charcoal specimen compared with the charcoal made at present-day of equal mass? Approximately $52 \%$ Approximately $63 \%$ Approximately $58 \%$ Approximately $55 \%$

Let $A(x, y, z)=x^{2}+x+y^{2}-2 y-z^{2}$ (a) Find the global maximum and minimum of $A(x, y, z)$ subject to the constraint $x^{2}+y^{2}+z^{2}=2$ (b) Find the global maximum and minimum of $A(x, y, z)$ on the closed bounded domain $x^{2}+y^{2}+z^{2} \leq 16$.

(1 point) a. Find the Laplace transform $F(s)=\mathcal{L}\{f(t)\}$ of the function $f(t)=\sin ^{2}(w t)$, defined on the interval $t \geq 0$. \[ F(s)=\mathcal{L}\left\{\sin ^{2}(w t)\right\}= \] \# help (formulas) Hint: Use a double-angle trigonometric identity. b. For what values of $s$ does the Laplace transform exist? \# help (inequalities)

Problem \#7: A sample of tritium-3 decayed to $83 \%$ of its original amount after 4 years. How long would it take the sample (in years) to decay to $23 \%$ of its original amount? Problem \#7: Enter your answer symbolically, as in these examples Just Save Submit Problem \#7 for Grading \begin{tabular}{|r|l|l|l|} \hline $\begin{array}{r}\text { Problem \#7 } \\ \text { Your Answer: } \\ \text { Your Mark: }\end{array}$ & $\underline{\text { Attempt \#1 }}$ & Attempt \#2 & Attempt \#3 \\ \hline \end{tabular}

Let $g$ be the function given by $g(x)=x^{2} e^{k x}$, where $k$ is a constant. For what value of $k$ does $g$ have a critical point at $x=\frac{2}{3}$.

1) Use technology to graph the direction field for the differential equation $\frac{d y}{d x}=1-x y$. Then, sketch by hand an approximate solution curve that passes through the given points. Copypaste the direction field here. a) $y(0)=0$ b) $y(-1)=0$ c) $y(2)=2$ d) $y(0)=-4$

A rectangular sheet of cardboard of size 45 cm by 21 cm is made into an open box by cutting away squares of equal size from each corner and folding up the sides. Let Acm3 be the largest volume of such a box that can be made in this way. Determine the value of A. ONLY answer if you have actually done the question. Do not copy paste other people answer thank you. I will downvote if i get any answers that don&qpos;t answer question. Thank you. 4. A rectangular sheet of cardboard of size $45 \mathrm{~cm}$ by $21 \mathrm{~cm}$ is made into an open box by cutting away squares of equal size from each corner and folding up the sides. Let $A \mathrm{~cm}^{3}$ be the largest volume of such a box that can be made in this way. Determine the value of $A$.

(1) (Inverse Functions) A boat sails directly away from a 250 meter tall skyscraper that stands on the edge of a harbor. Let $x$ be the horizontal distance between the base of the building and the boat. The angle $\theta$, measured in radians, is the angle of elevation from the boat to the top of the building. (a) Sketch a picture of this situation. (b) Give a formula relating the angle $\theta$ to the horizontal distance $x$ between the boat and the building. (c) Use your equation to solve for $\theta$. (d) What are the units of $\frac{d \theta}{d x}$ ? (e) Do you expect the value of $\frac{d \theta}{d x}$ to be positive or negative? Explain. (f) How fast is the angle of elevation changing when the boat is 100 meters from the building?

5. [0/4 Points] DETAILS MY NOTES Evaluate the integral by making the given substitution, giving your answer in terms of $x$. (Use c for the constant of integration.) \[ \int \cos (3 x) d x, \quad u=3 x \] 6. $[0 / 2$ Points $]$ DETAILS MY NOTES Evaluate the definite integral. \[ \int_{0}^{1} \sqrt{1+8 x} d x \]

Problem \# 8: Find the point on the following surface that has a positive $x$-coordinate and at which the tangent plane is parallel to the given plane. \[ x^{2}+3 y^{2}+10 z^{2}=21400 ; \quad 14 x+30 y+60 z=5 \] Problem \#8:

The following table shows the $x$ - and $y$-coordinates of five points that lie on a curve $y=f(x)$. \begin{tabular}{|c|c|c|c|c|c|} \hline$x$ & 2 & 2.5 & 3 & 3.5 & 4 \\ \hline$y=f(x)$ & 6 & 4.875 & 4 & 4.125 & 6 \\ \hline \end{tabular} (a) Estimate the area under the curve in the interval $2<x<4$. Use the trapezoidal rule with 4 partitions of equal width. The equation of the curve was found to be $y=x^{3}-7 x^{2}+14 x-2$. (b) (i) Write down an integral that represents the area under the curve over the interval. (ii) Find the exact value of the area under the curve over the interval. (c) Find the percentage error between the estimation in part (a) and the exact value in part (b). Provide a reason for the difference.

The temperature of Earth s surface is strongly influenced by the amount of carbon in the atmosphere. Each year, human activity adds another 9 gigatons of carbon to the atmosphere. 2 gigatons of this carbon is absorbed by the oceans, and 3 gigatons of this carbon is absorbed by plant growth (photosynthesis). There is currently 900 gigatons of carbon in the atmosphere. a) Assume that these are the only flows of carbon into/out of the atmosphere, and that the given rates stay constant over time. Write down a differential equation for the total amount of carbon in the atmosphere over time. Solve the equation and use the solution to predict how much carbon there will be in the atmosphere in 25 years. (b) Let us make our a bit more model realistic. Instead of assuming a constant rate of absorption by plant growth, assume that the rate of carbon absorption by plant growth is proportional to the amount of carbon in the atmosphere at any time t. This increase constitutes 0.4% for each gigaton of carbon. Write down a differential equation for the total amount of carbon in the atmosphere over time. Solve the equation and use the solution to predict the equilibrium amount of carbon in the atmosphere. (c) Now expand the model again and imagine that humans start to reduce their carbon emissions such that the rate at which carbon is added to the atmosphere decreases over time. Assume that this decreasing rate is described by the function h(t) = 7e-at +2; where a > 0 is a constant. d) Again write out and solve the differential equation describing the amount of carbon in the atmosphere. Assuming that a = 0.002, calculate how long it will take for the amount of atmospheric carbon to reach it maximum value. What is this maximum value?

Chance at time $t=0$, and Mary remains ahead of Chance for $0<t \leq 20$. (b) At time $t=10$, is Mary speeding up or slowing down? Give a reason for your answer. 0 / 10000 Word Limit (c) Is the distance between Mary and Chance at time $t=18$ increasing or decreasing? Give a reason for your answer

(1 point) A certain bacteria population P obeys the exponential growth law \[ P(t)=500 e^{2.9 t} \] (t in hours) (a) How many bacteria are present initially? (b) At what time will there be 10000 bacteria? (a) (b)

Can you answer all the questions mark in red including the last one please, thanks Suppose you are an industrial designer working for an automotive company, you are given a car deisgn to work on: with a black flat body and a red parabolic fender. The design profile is specified as a piecewise equation \[ f(x)=\left\{\begin{array}{cc} 1 & 0 \leq x \leq 1 \\ a x^{2}+b x+c & 1<x \leq 2 \end{array}\right. \] Your first job is to make the two pieces fit together. Have a go now with the $a, b$ and $c$ sliders. It's not an easy job! Mathematically, making the pieces fit together corresponds to the mathematical concept of $f$ being continuous at the point $x=1$ : \[ \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)=f(1) . \] Solving this equation gives you a condition on the coefficients $a, b$ and $c$, namely that Your second job is to make the two pieces fit together smoothly. This corresponds to the mathematical concept of $f$ being differentiable at the point $x=1$ : \[ \lim _{x \rightarrow 1^{-}} f^{\prime}(x)=\lim _{x \rightarrow 1^{+}} f^{\prime}(x) \] Solving this equation gives another condition on the coefficients $a, b$ and $c$, namely that So there is some freedom here. For example if you choose $a=-3$, then $b=$ and $c=$ Note: the Maple syntax for the condition $a+b c=0$ is $\mathrm{a}+\mathrm{b}^{\star} \mathrm{C}=0$. Let's find values of $a$ and $b$ such that the piecewise defined function \[ f(x)=\left\{\begin{array}{ll} x^{3}+3 & x \leq 1 \\ a x+b & x>1 \end{array}\right. \] is differentiable. First we make the function continuous: - $\lim _{x \rightarrow 1^{-}} f(x)=$ 国这 - $\lim _{x \rightarrow 1^{+}} f(x)=$ 国国 - $f(1)=$ 因至. These three quantities are equal if $a$ and $b$ satisfy the condition:

Consider ?=ℎ(?,?) as a parametrized surface in the natural way. Write the equation of the tangent plane to the surface at the point (−4,4,2) given that ∂ℎ/∂?(4,2)=3 and ∂ℎ/∂?(4,2)=0.

The position of an object is given by s ( t ) = 8 t + 10 sin ( t ) determine where in the interval [ 0,12] the object is moving to the right and moving to the left.

Question 3 The following graph gives the position, $f(t)$, of a particle at time $t$. Select each of the marked values of $t$ for which the statements could be true. $\boldsymbol{x}$ Incorrect. (c) The acceleration is positive. (e) The velocity is decreasing.

Question 7 (1 point) For what value of $k$, if any, will $y=k \sin (5 x)+2 \cos (4 x)$ be a solution to the differential equation $y^{\prime \prime}+16 y=-27 \sin (5 x)$ ? a $\quad-27$ b $-9 / 5$ c 3 d There is no such value of $k$multiple choice

(a) Assuming $n \geq 2$, find the indicated elements of the recurrence relation below: \[ \int_{0}^{x} \cos ^{n}(4 t) d t=F_{n}(x)+K_{n} \int_{0}^{x} \cos ^{n-2}(4 t) d t, \quad x \in \mathbb{R} . \] Answers: \[ \begin{array}{l} F_{n}(x)= \\ K_{n}= \end{array} \] (a) Assuming $n \geq 2$, find the indicated elements of the recurrence relation below: \[ \int_{0}^{x} \cos ^{n}(4 t) d t=F_{n}(x)+K_{n} \int_{0}^{x} \cos ^{n-2}(4 t) d t, \quad x \in \mathbb{R} . \] Answers: \[ \begin{array}{l} F_{n}(x)= \\ K_{n}= \end{array} \]

One method for determining the age of ancient objects is by radiocarbon dating. Natural processes convert nitrogen to Carbon-14, a radioactive isotope with a half-life of 5730 years. When they are alive, animals assimilate Carbon-14 through the food chain. When an animal dies, it stops replacing its carbon and the amount of Carbon-14 decays exponentially. (a) Suppose the minimum detectable amount is 0.1% of the initial amount. What is the maximum age of a fossil that we could date using Carbon-14? (b) Suppose we wanted to date a fossil that was potentially 70 million years old. A radioactive isotope with what half-life would be required?

(1 point) Given that $f(0.3)=5.1$ and $f(0.7)=-4.4$, approximate $f^{\prime}(0.3)$. \[ f^{\prime}(0.3) \approx 5 \]

Consider the logistic population growth given by $\frac{d P}{d t}=r P\left(1-\frac{P}{\kappa}\right)$. (a) This equation is of the form $d P / d t=f(P)$. Plot the function $f(P)$ vs $P$, classify the equilibria and sketch several graphs of solutions in the $t-P$ plane. (b) Find the solution if $0<P(0)<\kappa$ and check that the limiting behaviour agrees with your sketch in $(a)$. (c) Find the solution if $P(0)=\kappa$.

\begin{tabular}{|c|c|c|c|c|c|} \hline$x$ & 0 & 1 & 2 & 3 & 4 \\ \hline$g(x)$ & 1 & $\frac{1}{8}$ & $-\frac{3}{4}$ & $-\frac{13}{8}$ & $-\frac{5}{2}$ \\ \hline \end{tabular} Selected values of the twice-differentiable function $g$ are given in the table above. What is the value of $\int_{0}^{4} g^{\prime}(x) \arctan ^{2}(2 g(x)+3) d^{\prime} x$ ? (A) $-\mathbf{1 4 . 5 8 8}$ (B) -3.647 (C) -0.721 (D) 2.136

For the functions $w=x y+y z+x z, x=2 u+3 v, y=2 u-3 v$, and $z=u v$, express $\frac{\partial w}{\partial u}$ and $\frac{\partial w}{\partial v}$ using the chain rule and by expressing $w$ directly in terms of $u$ and $v$ before differentiating. Then evaluate $\frac{\partial w}{\partial u}$ and $\frac{\partial w}{\partial v}$ at the point $(u, v)=\left(-\frac{2}{3}, 3\right)$. Express $\frac{\partial w}{\partial \mathrm{u}}$ and $\frac{\partial w}{\partial \mathrm{v}}$ as functions of $u$ and $v$ \[ \frac{\partial w}{\partial u}= \]