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John is studying on some species of sharks and their interactions with their prey species, which mostly consists of sardine fish and some crabs. Therefore, his work falls within: Autecology Landscape ecology Synecology Spatial ecology

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\frac{dy}{dx} = 2x^2 + \frac{x^2 - 1}{x + 1}

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Question 6 (10 points) Refer to section 5 in the course notes. The graph below is of the function: $y = 9 - x^2$. Compute the shaded area and enter your answer as a decimal rounded to 2 places. Your Answer: Answer Question 7 (35 points) Answer all 3 questions, showing your work and simplifying. Upload a pdf or 1 page picture. 1) Refer to sections 1,2,3 in the course notes. Find the antiderivative of $4x^7 - 6x + 5$ 2) Refer to sections 1,2,3 in the course notes. Compute the integral: $\int (3e^t - \frac{4}{t}) dt$ 3) Refer to section 3 in the course notes. Find a function g(x) with $g'(x) = 8x^3 - 9x^2 + 4$ and $g(1) = -2.$

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Question 1 (5 points) (a) (2 points) Given r(t) = (f(t), g(t)), find the normal to the curve in a right-hand sense. (b) (3 points) Given r(t) = ($e^t$ cos t, $e^t$ sin t, $e^t$), find the integral of f(x, y, z) = $x^2$ + $y^2$ + $z^2$ along r(t) for 0 ? t ? 2?. Hint: we calculated ||dr/dt|| for this curve in Lecture 2, you may use that result here.

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In cell B9, enter a formula to calculate the average sales per month for the truck locations. Hint: Use cells B7:E7 from the MobileSales worksheet as the function argument.

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Find the area of the surface generated when the given curve is revolved about the given axis.\\ $y = \frac{x^3}{15}$, for $0 \leq x \leq \sqrt[3]{15}$; about the x-axis\\ The surface area is square units.\\ (Type an exact answer, using $\pi$ as needed.)

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6. If $\vec{i}$, $\vec{j}$, and $\vec{k}$ are the standard unit basic vectors in 3-space, determine the value of $\vec{k} \cdot (\vec{j} - 3\vec{k}) + (\vec{i} - 4\vec{k}) \cdot (\vec{i} - 4\vec{k}) - 8|\vec{i} \times \vec{k}|$. (3 marks) (Simplify without using components).

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5. A 5 V Zener diode is employed in the following circuit as a voltage limiter: $V_{in} = +16 V$ R1 ZD1 $+$ $V_o = +5V$ The Zener diode can safely dissipate 1 W at temperatures < 50 C, and is derated at 6.67 mW/C. If the ambient temperature is equal to 100 C, what is the minimum permissible value of R1 at this temperature ? (8 marks)

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#2. A schematic of a spherical medication tablet is shown to the right. The active ingredient (naproxen, found in Aleve) diffuses through the three layers shown, which are designed to provide a more uniform release of the drug out of the tablet. The thickness of the layers 1, 2, and 3 and their mass transfer properties are shown in the table below. The surface drug concentration just outside layer 1 is 1.062 mg/mm$^3$. Assume the concentration just outside layer 3 is zero. Calculate the steady state rate of naproxen leaving the outer surface under these conditions. The diffusion mass transfer resistance equation for a spherical shell can be found in Table VIII-8. (0.41 mg/s) Layer Thickness (mm) Diffusion Coefficient (of drug in layer) (mm$^2$/s) Partition Coefficient ($K_{layer/solution}$) 1 (Methocel K100) 0.89 $9.7 \times 10^{-2}$ 0.5 2 (Methocel E10) 0.76 $0.15 \times 10^{-2}$ 0.5 3 (PVA) 0.13 $0.24 \times 10^{-2}$ 0.7

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Laura leased equipment worth $70,000 for 7 years. If the cost of borrowing is 4.76% compounded monthly, calculate the size of the lease payment that is required to be made at the beginning of each month.

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mackenzie m.