Calculus 1: midterm 2 solutions

This is for a test for Calculus 1: course.

Question 1

Calculate the derivative of $$ f(x) = x^{e} + e^{x} + x + e $$

---

$$\begin{aligned} f^{\prime} & = ( x^{e} + e^{x} + x + e)^{\prime} \\ & = (x^{e})^{\prime} + (e^{x})^{\prime} + (x)^{\prime} + (e)^{\prime} \\ & = ex^{e-1} + e^{x} + 1 + 0 \end{aligned}$$

Question 2

Differentiate $$ g(x) = \sqrt{x} \cos (x) $$

---

$$\begin{aligned} g^{\prime} & = (\sqrt{x} \cos x)^{\prime} \\ & = (\sqrt{x})^{\prime} \cos x + \sqrt{x} (\cos x)^{\prime} \\ &= \frac{1}{2\sqrt{x}} \cos x + \sqrt{x}(-\sin x) \end{aligned}$$

Question 3

Evaluate $\frac{dy}{dx}$ for $$ y = \sqrt{e^{x}} $$

---

$$\begin{alignat}{2} y & = \sqrt{u} & \quad u &= e^{x} \\ \frac{dy}{du} &= \frac{1}{2\sqrt{u}} & \quad \frac{du}{dx} &= e^{x} \end{alignat}$$ Therefore $$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx} = \frac{1}{2\sqrt{u}}e^{x} = \frac{1}{2\sqrt{e^{x}}}\cdot e^{x} $$

Question 4

Evaluate $\frac{dy}{dx}$ for $$xy = \cos y + x $$

---

$$\begin{aligned} \frac{d}{dx}(xy) &= \frac{d}{dx}(\cos y + x ) \\ \frac{d}{dx}(x) y + x\frac{d}{dx}(y) &= \frac{d}{dx}(\cos y) + \frac{d}{dx}(x) \\ 1\cdot y + x \cdot \frac{dy}{dx} & = -\sin y \frac{dy}{dx} + 1 \\ \frac{dy}{dx} & = \frac{1-y}{x + \sin y} \end{aligned}$$

Question 5

Suppose the altitude, in $m$, of an object is given by the function $$ y = t^{2} + t, \quad t \geq 0 $$ where $t$ is time, in sec. What is the velocity when the altitide is 12 meters?

---

Altitude $$ f(t) = t^{2} + t$$ Velocity $$f^{\prime}(t) = 2t + 1 $$ Altitude is 12 $$\therefore t^{2} + t = 12$$ Solve for $x$ $$ x= 3$$ Velocity $$f^{\prime}(3) = 2\cdot 3 + 1 = 7$$

Question 6

The population of a city declines by 10% every year. How long will it take to drop by 50% of the current population?

---

Population $$f(t) = Ce^{kt}$$ Declines by 10% in a year $$ 0.9 = 1 e^{k\cdot 1} $$ Solve for $k$ $$ k = \ln 0.9$$ Drops to 50% in $t$ years $$ 0.5 = 1e^{k\cdot t}$$ Solve for $t$ $$kt = \ln 0.5$$ So $$ t = \frac{\ln 0.5}{k} = \frac{\ln 0.5}{\ln 0.9} $$

Question 7

The area of a circle is increasing at a rate of 5 cm²/sec. At what rate is the radius of the circle increasing when the area is 2cm?

---

$r$ is the radius, $A$ is the area, $t$ is time. $$ A = \pi r^{2} $$ Differentiate with respect to $t$ $$\frac{dA}{dt} = \pi r \frac{dr}{dt} $$ Substitute $$ 5 = \pi\cdot 2 \frac{dr}{dt} $$ So $$\frac{dr}{dt} = \frac{5}{2\pi} $$

Question 8

Find the linear approximation of $f(x) = 3\sin(x)$ at $a = 0$. Use it to estimate $\sin(0.02)$.

---

$$\begin{aligned} f^{\prime}(x) &= 3\cos x \\ f^{\prime}(0) &= 3\cos 0 = 3 \end{aligned}$$ Then $$\begin{aligned} \ell(x) &= f(0) + f^{\prime}(0)( x - 0) \\ & = 0 + 3( x-0 ) \\ &= 3x \end{aligned}$$ So $$\ell(-0.02) = 3\cdot (-0.02) = -0.06$$