Calculus

Problem 9101

A rocket emits gases at $20.4 \mathrm{~m/s}$ , starts at $1.1 \mathrm{~kg}$ , and $g = 9.81 \mathrm{~m/s}^2$ .
(a) After $0.1$ s, it reaches $2.66 \mathrm{~m/s}$ . Find $\alpha$ (fuel burn rate in $\mathrm{kg/s}$ ) to 2 decimal places.
(b) How long until all fuel is used? Enter in seconds (2 decimal places).
(c) Assuming negligible shell mass, what height will the rocket reach when fuel is gone? Enter in metres (nearest metre).

See Solution

Problem 9102

Find the arc length $s$ of the curve $\vec{r}(t)=\langle 5 \sin (5 t), 5 \cos (5 t), 3 t\rangle$ for $-1 \leq t \leq 1$ .

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Problem 9103

Find the arc length $s$ of the curve $\vec{r}(t)=\langle-4 t+5,-4 t-2,-3 t-4\rangle$ for $-1 \leq t \leq 1$ .

See Solution

Problem 9104

Find the local extrema of $g(x, y)=x^{3}+22.5 x^{2}-1.5 y^{2}+150 x-18 y+70$ by calculating its partial derivatives.

See Solution

Problem 9105

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle-5 e^{3 t},-e^{3 t}, 2 e^{3 t}\rangle$ for $0 \leq t \leq \ln(4)$ . $s=$

See Solution

Problem 9106

Calculate the arclength of the curve $\vec{r}(t)=\left\langle 6 t, \frac{t^{3}}{3}, \frac{\sqrt{12} t^{2}}{2}\right\rangle$ for $-1 \leq t \leq 4$ .

See Solution

Problem 9107

Calculate the arclength of the curve $\vec{r}(t)=\left\langle 6 t, \frac{t^{3}}{3}, \frac{\sqrt{12} t^{2}}{2}\right\rangle$ for $-1 \leq t \leq 4$ .

See Solution

Problem 9108

Differentiate $y=\left(x^{2}+2\right)^{2}\left(x^{4}+4\right)^{4}$ using logarithmic differentiation.

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Problem 9109

Calculate the arclength of $\vec{r}(t)=\left\langle 3 t, \frac{t^{3}}{3}, \frac{\sqrt{6} t^{2}}{2}\right\rangle$ for $-3 \leq t \leq 1$ .

See Solution

Problem 9110

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle 12 \cos (-t), 2 \sqrt{13} \cos (-t), 14 \sin (-t)\rangle$ from $t=0$ to $t=\pi$ . $s=$

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Problem 9111

Find the derivative of $y=\frac{e^{-x} \cos ^{2} x}{x^{2}+x+1}$ using logarithmic differentiation.

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Problem 9112

Differentiate $y=x^{\sin x}$ using logarithmic differentiation.

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Problem 9113

Find the arc length function $s(t)$ and the arc length parameterization $\vec{r}(s)$ for the curve $\vec{r}(t)=\langle\cos (t), \sin (t), t\rangle$ .

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Problem 9114

Calculate the arclength of the curve $\vec{r}(t)=\left\langle 7 t, \frac{t^{3}}{3}, \frac{\sqrt{14} t^{2}}{2}\right\rangle$ for $t$ in $[-1, 4]$ .

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Problem 9115

Prove that $\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}$ .

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Problem 9116

Find the arc length function $s(t)$ and the arc length parameterization $\vec{r}(s)$ for the curve $\vec{r}(t)=\langle\cos (t), \sin (t), t\rangle$ .

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Problem 9117

Bestimmen Sie die Tangentengleichung an $K$ für $f(x)=x^{2}-3x+2$ im Punkt P $(4 \mid f(4))$ .

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Problem 9118

Find the arc length function $s(t)$ for the curve $\vec{r}(t)=\langle\cos(t), \sin(t), t\rangle$ and its arc length parameterization.

See Solution

Problem 9119

Find the derivative of $f(x)=\frac{x^{5}-x}{x^{3}}$ .

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Problem 9120

Calculate the arclength of the curve $\vec{r}(t)=\left\langle 6 t, \frac{t^{3}}{3}, \frac{\sqrt{12} t^{2}}{2}\right\rangle$ for $-2 \leq t \leq 3$ .

See Solution

Problem 9121

Find the derivative of $f(x)=x \cdot \sin(x)$ using the product rule.

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Problem 9122

Bestimme die Tangentengleichung an $K$ von $f$ im Punkt $A(-2 \mid \ldots)$ für $f(x)=\frac{1}{4} x^{4}-\frac{3}{2} x^{2}$ .

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Problem 9123

A fish is reeled in at 3.7 ft/s from 10 ft above water. Find the rate of angle change when 25 ft of line is out. $\frac{d \theta}{d t}=$ $\mathrm{rad} / \mathrm{sec}$

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Problem 9124

Find the following for the function $f(x)=5 x^{2}-9$ : a. $\frac{f(x)-f(a)}{x-a}$ ; b. $\frac{f(x+h)-f(x)}{h}$ .

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Problem 9125

Find the curvature of the curve $\vec{r}(t)=\langle t-\sin t, 1-\cos t\rangle$ at $t=\frac{7 \pi}{4}$ .

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Problem 9126

Find the curvature of the curve $\vec{r}(t)=\langle 3t, -4-3t, t \rangle$ at $t=-5$ .

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Problem 9127

Find the arc length function $s(t)$ and the arc length parameterization $\vec{r}(s)$ for the curve $\vec{r}(t)=\langle 5 \cos (t), 7 \sin (t), 2 \sqrt{6} \cos (t)\rangle$ .

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Problem 9128

Find the curvature of the curve $\vec{r}(t)=\langle 4 \cos (3 t), 4 \sin (3 t), t\rangle$ at $t=0$ , rounded to two decimal places.

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Problem 9129

Find the arc length function $s(t)$ and the arc length parameterization $\vec{r}(s)$ for the curve $\vec{r}(t)=\langle-4 t+3,3 t-3,-5 t+1\rangle$ .

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Problem 9130

Find the curvature of the curve $\vec{r}(t)=\langle -t, 4t^4, -3t^2 \rangle$ at $t=-2$ . Round to 2 decimal places.

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Problem 9131

Find the following for the function $f(x)=4x+8$ : a. $\frac{f(x)-f(a)}{x-a}$ , b. $\frac{f(x+h)-f(x)}{h}$ .

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Problem 9132

Determine which function has a horizontal asymptote at $y=0$ : $f(x)=\frac{x^{2}+2}{x-1}$ or $g(x)=e^{x}$ . Choose (A), (B), (C), or (D).

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Problem 9133

Calculate the curvature of the curve $\vec{r}(t)=\langle 1-3 \cos t, 0,4+3 \sin t\rangle$ at $t=\frac{1}{4} \pi$ .

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Problem 9134

Gegeben ist die Funktion $f(x)=\frac{1}{4} x^{2}(x-3)$ .
a) Zeigen Sie, dass die Tangente $t$ an $K$ bei $x=1$ durch $P(3 \mid-2)$ verläuft. b) Bestimmen Sie die Tangenten an $K$ , die parallel zu $y=2,25 x-1$ sind. c) Wo hat $K$ eine waagrechte Tangente? d) Wo verläuft die Tangente an $K$ parallel zur Geraden mit Steigung $m=6$ ?

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Problem 9135

Find the leading term in the Taylor series of $f(x)=\int_{0}^{x} \ln (\cosh (t)) dt$ around zero.

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Problem 9136

Find the derivatives of these functions:
1. $f(x)=-2 x^{4}+3 x^{2}-4 x+2$
2. $f(x)=\frac{1}{4} x^{4}+\frac{3}{8} x^{3}-\frac{4}{5} x^{2}$
3. $f(x)=0.5 x^{4}-x^{3}+2.5 x^{2}-8$
4. $f(x)=-x^{4}-6 x^{2}-2.25$
5. $f(x)=\frac{1}{32} x^{3}+\frac{3}{7} x-\frac{4}{x}$
6. $f(x)=-(x-6)^{2}(x+1)$
7. $f(x)=-\frac{5}{6} x^{2}+\frac{2}{3} x+\frac{5}{2}$
8. $f(x)=\frac{1}{16}(x^{5}+x^{3}-1)$
9. $f(x)=0.5(x^{2}-2)^{2}$
10. $I(t)=0.125 t^{4}-1.5 t^{2}-3$
11. $A(u)=u(u^{2}-1.5 u-4)$
12. $f(x)=a x^{2}+b x+c+\frac{d}{x}$
13. $f(x)=\frac{1}{8}(x-1)(x^{2}-12 x+16)$
14. $k(x)=a x^{3}+b x^{2}+c x+d$

See Solution

Problem 9137

Find the Tangent $\vec{T}$ , Normal $\vec{N}$ , and Binormal $\vec{B}$ vectors for $\vec{r}(t)=\langle 3 \cos (2 t), 3 \sin (2 t), 4 t\rangle$ at $t=0$ .

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Problem 9138

Find the Tangent, Normal, and Binormal vectors $\vec{T}$ , $\vec{N}$ , and $\vec{B}$ for $\vec{r}(t)=\langle 4 \cos (2 t), 4 \sin (2 t), 4 t\rangle$ at $t=0$ .

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Problem 9139

Find the Tangent, Normal, and Binormal vectors $\vec{T}$ , $\vec{N}$ , and $\vec{B}$ for $\vec{r}(t)=\langle 3 \cos (2 t), 3 \sin (2 t), 4 t\rangle$ at $t=0$ .

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Problem 9140

Find the curvature of the curve $\vec{r}(t)=\langle\cos (2 t), \sin (2 t), t\rangle$ at $t=0$ . Round to two decimal places.

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Problem 9141

Evaluate the integral: $I = \frac{1}{2} \int \sin(2\theta) \sin(\theta) d\theta$ .

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Problem 9142

Find the velocity and acceleration vectors for $\vec{r}(t)=\langle-2 t, 3 t^{4}, 5 t^{3}+5\rangle$ at $t=-1$ .

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Problem 9143

Determine the acceleration vector for $\vec{r}(t)=\left\langle\sin (6 t), 11 t^{5}, e^{-3 t}\right\rangle$ . Find each component.

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Problem 9144

Find the velocity vector for $\vec{r}(t)=\langle\sin(8t), 10t^{10}, e^{-4t}\rangle$ . Compute each component.

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Problem 9145

Find the speed of the object with position $\bar{r}(t)=\langle 7 t^{2}+7,4 t-6,t^{3}+7 t\rangle$ at $t=4$ . Show answer to 4 decimal places. Speed =

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Problem 9146

Find the tangent line equation for the function $f(x)=\frac{x-13}{5x+6}$ at $x=3$ . Tangent line: $y=$

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Problem 9147

Gegeben ist die Funktion $f$ mit $f(x)=x^{2}-2$ .
a) Finde die Tangentengleichung an $P(0,5 \mid f(0,5)$ ). b) Bestimme Punkte mit Steigung 4 und 0. c) Wo ist die Tangente parallel zu $y=-2x+3$ ?

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Problem 9148

Find the position vector $\vec{r}(t)$ for a particle with acceleration $\vec{a}(t)=\langle 2 t, 5 \sin (t), \cos (4 t)\rangle$ , initial velocity $\vec{v}(0)=\langle 3,1,0\rangle$ , and initial position $\vec{r}(0)=\langle-2,-5,-1\rangle$ .

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Problem 9149

Approximate $\sqrt{25.3}$ using the linear approximation of $f(x)=\sqrt{x}$ at $x=25$ . Find $m$ and $b$ for $L(x)=m x+b$ .

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Problem 9150

Find the horizontal velocity, vertical velocity, and speed of an object at $t=2$ given $x=4t^{2}+3t$ , $y=5t^{2}+1$ .

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Problem 9151

Find the derivative $Y'(u)$ of the function $Y(u)=(u^{-2}+u^{-3})(u^{5}+5u^{2})$ .

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Problem 9152

Find points on the curve where the tangent line is parallel to $y=-2x$ for the equation $2x+y+x \frac{dy}{dx}+2y \frac{dy}{dx}=0$ .

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Problem 9153

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x}$

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Problem 9154

Find the degree 3 MacLaurin polynomial for $f(x)=e^{2x}$ and use it to approximate $e^{6}$ . $T_{3}(x)=\square+\square x+\square x^{2}+\square x^{3}$ $e^{6} \approx$

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Problem 9155

Find the area between $f(x)=e^{x} \sec ^{2} x$ and $g(x)=e^{x} \tan ^{2} x$ from $0$ to $\pi$ , avoiding singularities.

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Problem 9156

Find the tangential and normal components of the acceleration vector for $\vec{r}(t)=\langle-3t, t^5, t^2\rangle$ at $t=2$ .

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Problem 9157

Given $f^{\prime}(2)=4$ and $g^{\prime}(2)=-7$ , find $h^{\prime}(2)$ for: (A) $h(x)=6 f(x)$ , (B) $h(x)=-13 g(x)$ , (C) $h(x)=12 f(x)+9 g(x)$ , (D) $h(x)=4 g(x)-7 f(x)$ , (E) $h(x)=4 f(x)+11 g(x)-2$ , (F) $h(x)=-6 g(x)-13 f(x)-4 x$ .

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Problem 9158

Given $f(1)=2$ and $f^{\prime}(1)=3$ , find $g^{\prime}(1)$ and $h^{\prime}(1)$ , then the tangent lines for $g(x)$ and $h(x)$ .

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Problem 9159

A baseball is hit from 4 feet high at 116 ft/s and $34^{\circ}$ . What is its max height? Neglect air resistance.

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Problem 9160

Gegeben ist $Q(0)=1 \mathrm{As}$ .
a) Finde $Q(t)$ allgemein.
b) Berechne $Q(t)$ für $I_{0}=1 \mathrm{A}$ und $t_{1}, t_{2}, t_{3}$ .
c) Zeichne den Ladungsverlauf.

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Problem 9161

Gegeben ist ein Stromimpuls mit $Q(0)=Q_{0}=1$ As. Berechnen Sie $Q(t)$ allgemein und für $I_{0}=1 \mathrm{~A}$ , $t_{1}=1 \mathrm{~s}$ , $t_{2}=2 \mathrm{~s}$ , $t_{3}=3 \mathrm{~s}$ . Stellen Sie $Q(t)$ grafisch dar.

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Problem 9162

Find dy/dt for y=(1+sin(8t))^{-6} when g=3.6. Show your work.

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Problem 9163

Peter has 60 feet of fencing for 2 adjacent rabbit pens. What dimensions maximize their area?

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Problem 9164

Find the second derivative $y^{\prime \prime}$ for the function $y=\sqrt{9 x+10}$ . Show your work.

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Problem 9165

Find the derivatives of the following functions with constant $a$ : (a) $y=x^{a}$ , (b) $y=a^{x}$ , (c) $y=x^{x}$ , (d) $y=a^{a}$ .

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Problem 9166

Find the derivative of $y$ with respect to $x$ for the function $y=4^{x}$ .

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Problem 9167

Differentiate the function: $\ln \left(3 x^{5} e^{2}\right)$ .

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Problem 9168

Evaluate the infinite series: $\sum_{n=1}^{\infty} \frac{1+3^{n}}{3^{n}+2^{n}}$ .

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Problem 9169

Find the derivatives and simplify: (a) $\frac{d}{d x}\left[e^{2 \ln x}\right]$ (b) $\frac{d}{d x}\left[\log _{a} a^{\sin x}\right]$ (c) $\frac{d}{d x}\left[\log _{2} 8^{x-5}\right]$

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Problem 9170

At noon, ship A is 180 km west of ship B. A sails south at 20 km/h, B sails north at 40 km/h. What's the distance change rate at 4 PM? $\square$ km/h

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Problem 9171

Find the derivative of $y$ for $y=7^{\sqrt{t}}$ with respect to $t$ .

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Problem 9172

What is the speed of a ball dropped from a height of $15 \mathrm{~m}$ when it hits the ground?

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Problem 9173

Find $dy/dx$ using implicit differentiation for the equation $2xy - y^2 = 1$ . Show your work.

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Problem 9174

Find the derivative of the function $f(x)=x^{2} \cdot \arctan (5 x)$ .

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Problem 9175

Find the surface area of a parabolic dish antenna modeled by $y=\sqrt{\frac{K}{4}} x^{2}$ , $0 \leq x \leq R$ .

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Problem 9176

Find the arc length of the curve defined by $x=t^{3}$ and $y=t^{2}$ from $t=-1$ to $t=1$ . Round to two decimal places.

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Problem 9177

Find the tangent line equation to the curve $y^{6}+x^{3}=y^{2}+12 x$ at the point $(0, 1)$ .

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Problem 9178

Find the limit: $\lim _{x \rightarrow 0^{+}} 8 \sin (x) \ln (x)$ . Use l'Hopital's rule if needed.

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Problem 9179

Find the limit using L'Hospital's rule: $\lim _{x \rightarrow 0} \frac{\sin (7 x)}{\tan (8 x)}$ .

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Problem 9180

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{14 e^{x}-14 x-14}{13 x^{2}}=$ . Enter INF, -INF, or DNE.

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Problem 9181

Find all values of $c$ in $(-3, 3)$ such that $f'(c) = \frac{f(3) - f(-3)}{3 - (-3)}$ for $f(x) = x^3 - 7x + 4$ .

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Problem 9182

Estimate $f(7) - f(4)$ using the Mean Value Theorem, given $-3 \leq f'(x) \leq 2$ on $(4,7)$ .

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Problem 9183

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0^{+}}(1+4 x)^{3 / x}=$ Enter INF, -INF, or DNE.

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Problem 9184

A camera is $4000 \mathrm{~m}$ from a rocket. When the rocket rises $3000 \mathrm{~m}$ at $600 \mathrm{~m/s}$ , find: (a) distance change rate, (b) angle change rate. (a) $\square$ m/s (b) $\square$ radians/s

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Problem 9185

Identify which limits are indeterminate forms given the limits as $x$ approaches $a$ :
(a) $\lim _{x \rightarrow a}[f(x)-p(x)]=$ (b) $\lim _{x \rightarrow a}[p(x)-q(x)]=$ (c) $\lim _{x \rightarrow a}[p(x)+q(x)]=$
Output I, INF, NINF, or D for each.

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Problem 9186

A bacteria population starts at 3000 and grows to 9000 in 34 hours. Find the growth constant $k$ .

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Problem 9187

A cone-shaped tank has a top radius of $3 m$ and height $5 m$ . Water is pumped in at $2 \mathrm{~m}^{3} / \mathrm{min}$ . Find the rise rate of water level when it's $2 m$ deep in $\mathrm{m} / \mathrm{min}$ .

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Problem 9188

Find the population growth rate in people per year for $P(t)=2000 e^{0.05 t}$ at $t=0$ .

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Problem 9189

A square's side length decreases at 2 km/h. If it's 9 km now, find the area change rate in km²/h.

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Problem 9190

Find the derivative of $y$ where $y = \cos^{-1}(3x^9)$ .

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Problem 9191

Find $\frac{dy}{dt}$ for $y=\arcsin (\sqrt{11} t)$ .

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Problem 9192

Find the speed of an object dropped from a 625 m cliff after 4 seconds, where height is $625 - 4.9 t^{2}$ .

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Problem 9193

Analyze the behavior of $f(t)=-4(4)^{t}+4$ as $t \to -\infty$ and $t \to \infty$ . What do $f(t)$ approach?

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Problem 9194

32

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Problem 9195

Evaluate the double integral $\iint_{\mathbf{R}} x \cos (2 x+y) d A$ over $0 \leq x \leq \frac{\pi}{3}, 0 \leq y \leq \frac{\pi}{4}$ .

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Problem 9196

Given Boyle's Law $PV=C$ , find the rate of volume decrease when $V=375 \mathrm{~cm}^{3}$ , $P=140 \mathrm{kPa}$ , and $\frac{dP}{dt}=16 \mathrm{kPa/min}$ .

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Problem 9197

Find the average rate of change of $h(x)=x^{2}-7x+9$ from $x=-1$ to $x=10$ .

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Problem 9198

Evaluate the differential $dy$ for $y=x^{2}+4x$ at $x=2$ and $dx=\frac{1}{2}$ . Provide only the final answer.

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Problem 9199

Find the limit using I'Hospital's Rule: $\lim _{x \rightarrow 1} \frac{6 \ln x}{7 \sin \pi x}$ .

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Problem 9200

Find the $x$ in $y=3x^2+2x+1$ on $[-2,1]$ where the derivative equals the slope of the secant line.

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Calculus

Problem 9101

Problem 9102

Find the arc length sss of the curve r⃗(t)=⟨5sin⁡(5t),5cos⁡(5t),3t⟩\vec{r}(t)=\langle 5 \sin (5 t), 5 \cos (5 t), 3 t\rangler(t)=⟨5sin(5t),5cos(5t),3t⟩ for −1≤t≤1-1 \leq t \leq 1−1≤t≤1.

Problem 9103

Find the arc length sss of the curve r⃗(t)=⟨−4t+5,−4t−2,−3t−4⟩\vec{r}(t)=\langle-4 t+5,-4 t-2,-3 t-4\rangler(t)=⟨−4t+5,−4t−2,−3t−4⟩ for −1≤t≤1-1 \leq t \leq 1−1≤t≤1.

Problem 9104

Find the local extrema of g(x,y)=x3+22.5x2−1.5y2+150x−18y+70g(x, y)=x^{3}+22.5 x^{2}-1.5 y^{2}+150 x-18 y+70g(x,y)=x3+22.5x2−1.5y2+150x−18y+70 by calculating its partial derivatives.

Problem 9105

Calculate the arc length sss of the curve r⃗(t)=⟨−5e3t,−e3t,2e3t⟩\vec{r}(t)=\langle-5 e^{3 t},-e^{3 t}, 2 e^{3 t}\rangler(t)=⟨−5e3t,−e3t,2e3t⟩ for 0≤t≤ln⁡(4)0 \leq t \leq \ln(4)0≤t≤ln(4). s=s= s=

Problem 9106

Calculate the arclength of the curve r⃗(t)=⟨6t,t33,12t22⟩\vec{r}(t)=\left\langle 6 t, \frac{t^{3}}{3}, \frac{\sqrt{12} t^{2}}{2}\right\rangler(t)=⟨6t,3t3​,212​t2​⟩ for −1≤t≤4-1 \leq t \leq 4−1≤t≤4.

Problem 9107

Calculate the arclength of the curve r⃗(t)=⟨6t,t33,12t22⟩\vec{r}(t)=\left\langle 6 t, \frac{t^{3}}{3}, \frac{\sqrt{12} t^{2}}{2}\right\rangler(t)=⟨6t,3t3​,212​t2​⟩ for −1≤t≤4-1 \leq t \leq 4−1≤t≤4.

Problem 9108

Differentiate y=(x2+2)2(x4+4)4y=\left(x^{2}+2\right)^{2}\left(x^{4}+4\right)^{4}y=(x2+2)2(x4+4)4 using logarithmic differentiation.

Problem 9109

Calculate the arclength of r⃗(t)=⟨3t,t33,6t22⟩\vec{r}(t)=\left\langle 3 t, \frac{t^{3}}{3}, \frac{\sqrt{6} t^{2}}{2}\right\rangler(t)=⟨3t,3t3​,26​t2​⟩ for −3≤t≤1-3 \leq t \leq 1−3≤t≤1.

Problem 9110

Calculate the arc length sss of the curve r⃗(t)=⟨12cos⁡(−t),213cos⁡(−t),14sin⁡(−t)⟩\vec{r}(t)=\langle 12 \cos (-t), 2 \sqrt{13} \cos (-t), 14 \sin (-t)\rangler(t)=⟨12cos(−t),213​cos(−t),14sin(−t)⟩ from t=0t=0t=0 to t=πt=\pit=π. s=s= s=

Problem 9111

Find the derivative of y=e−xcos⁡2xx2+x+1y=\frac{e^{-x} \cos ^{2} x}{x^{2}+x+1}y=x2+x+1e−xcos2x​ using logarithmic differentiation.

Problem 9112

Differentiate y=xsin⁡xy=x^{\sin x}y=xsinx using logarithmic differentiation.

Problem 9113

Find the arc length function s(t)s(t)s(t) and the arc length parameterization r⃗(s)\vec{r}(s)r(s) for the curve r⃗(t)=⟨cos⁡(t),sin⁡(t),t⟩\vec{r}(t)=\langle\cos (t), \sin (t), t\rangler(t)=⟨cos(t),sin(t),t⟩.

Problem 9114

Calculate the arclength of the curve r⃗(t)=⟨7t,t33,14t22⟩\vec{r}(t)=\left\langle 7 t, \frac{t^{3}}{3}, \frac{\sqrt{14} t^{2}}{2}\right\rangler(t)=⟨7t,3t3​,214​t2​⟩ for ttt in [−1,4][-1, 4][−1,4].

Problem 9115

Prove that lim⁡n→∞(1+xn)n=ex\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}limn→∞​(1+nx​)n=ex.

Problem 9116

Find the arc length function s(t)s(t)s(t) and the arc length parameterization r⃗(s)\vec{r}(s)r(s) for the curve r⃗(t)=⟨cos⁡(t),sin⁡(t),t⟩\vec{r}(t)=\langle\cos (t), \sin (t), t\rangler(t)=⟨cos(t),sin(t),t⟩.

Problem 9117

Bestimmen Sie die Tangentengleichung an KKK für f(x)=x2−3x+2f(x)=x^{2}-3x+2f(x)=x2−3x+2 im Punkt P (4∣f(4))(4 \mid f(4))(4∣f(4)).

Problem 9118

Find the arc length function s(t)s(t)s(t) for the curve r⃗(t)=⟨cos⁡(t),sin⁡(t),t⟩\vec{r}(t)=\langle\cos(t), \sin(t), t\rangler(t)=⟨cos(t),sin(t),t⟩ and its arc length parameterization.

Problem 9119

Find the derivative of f(x)=x5−xx3f(x)=\frac{x^{5}-x}{x^{3}}f(x)=x3x5−x​.

Problem 9120

Calculate the arclength of the curve r⃗(t)=⟨6t,t33,12t22⟩\vec{r}(t)=\left\langle 6 t, \frac{t^{3}}{3}, \frac{\sqrt{12} t^{2}}{2}\right\rangler(t)=⟨6t,3t3​,212​t2​⟩ for −2≤t≤3-2 \leq t \leq 3−2≤t≤3.

Problem 9121

Find the derivative of f(x)=x⋅sin⁡(x)f(x)=x \cdot \sin(x)f(x)=x⋅sin(x) using the product rule.

Problem 9122

Bestimme die Tangentengleichung an KKK von fff im Punkt A(−2∣…)A(-2 \mid \ldots)A(−2∣…) für f(x)=14x4−32x2f(x)=\frac{1}{4} x^{4}-\frac{3}{2} x^{2}f(x)=41​x4−23​x2.

Problem 9123

A fish is reeled in at 3.7 ft/s from 10 ft above water. Find the rate of angle change when 25 ft of line is out. dθdt=\frac{d \theta}{d t}=dtdθ​= rad/sec\mathrm{rad} / \mathrm{sec}rad/sec

Problem 9124

Find the following for the function f(x)=5x2−9f(x)=5 x^{2}-9f(x)=5x2−9: a. f(x)−f(a)x−a\frac{f(x)-f(a)}{x-a}x−af(x)−f(a)​; b. f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​.

Problem 9125

Find the curvature of the curve r⃗(t)=⟨t−sin⁡t,1−cos⁡t⟩\vec{r}(t)=\langle t-\sin t, 1-\cos t\rangler(t)=⟨t−sint,1−cost⟩ at t=7π4t=\frac{7 \pi}{4}t=47π​.

Problem 9126

Find the curvature of the curve r⃗(t)=⟨3t,−4−3t,t⟩\vec{r}(t)=\langle 3t, -4-3t, t \rangler(t)=⟨3t,−4−3t,t⟩ at t=−5t=-5t=−5.

Problem 9127

Find the arc length function s(t)s(t)s(t) and the arc length parameterization r⃗(s)\vec{r}(s)r(s) for the curve r⃗(t)=⟨5cos⁡(t),7sin⁡(t),26cos⁡(t)⟩\vec{r}(t)=\langle 5 \cos (t), 7 \sin (t), 2 \sqrt{6} \cos (t)\rangler(t)=⟨5cos(t),7sin(t),26​cos(t)⟩.

Problem 9128

Find the curvature of the curve r⃗(t)=⟨4cos⁡(3t),4sin⁡(3t),t⟩\vec{r}(t)=\langle 4 \cos (3 t), 4 \sin (3 t), t\rangler(t)=⟨4cos(3t),4sin(3t),t⟩ at t=0t=0t=0, rounded to two decimal places.

Problem 9129

Find the arc length function s(t)s(t)s(t) and the arc length parameterization r⃗(s)\vec{r}(s)r(s) for the curve r⃗(t)=⟨−4t+3,3t−3,−5t+1⟩\vec{r}(t)=\langle-4 t+3,3 t-3,-5 t+1\rangler(t)=⟨−4t+3,3t−3,−5t+1⟩.

Problem 9130

Find the curvature of the curve r⃗(t)=⟨−t,4t4,−3t2⟩\vec{r}(t)=\langle -t, 4t^4, -3t^2 \rangler(t)=⟨−t,4t4,−3t2⟩ at t=−2t=-2t=−2. Round to 2 decimal places.

Problem 9131

Find the following for the function f(x)=4x+8f(x)=4x+8f(x)=4x+8: a. f(x)−f(a)x−a\frac{f(x)-f(a)}{x-a}x−af(x)−f(a)​, b. f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​.

Problem 9132

Determine which function has a horizontal asymptote at y=0y=0y=0: f(x)=x2+2x−1f(x)=\frac{x^{2}+2}{x-1}f(x)=x−1x2+2​ or g(x)=exg(x)=e^{x}g(x)=ex. Choose (A), (B), (C), or (D).

Problem 9133

Calculate the curvature of the curve r⃗(t)=⟨1−3cos⁡t,0,4+3sin⁡t⟩\vec{r}(t)=\langle 1-3 \cos t, 0,4+3 \sin t\rangler(t)=⟨1−3cost,0,4+3sint⟩ at t=14πt=\frac{1}{4} \pit=41​π.

Problem 9134

Problem 9135

Find the leading term in the Taylor series of f(x)=∫0xln⁡(cosh⁡(t))dtf(x)=\int_{0}^{x} \ln (\cosh (t)) dtf(x)=∫0x​ln(cosh(t))dt around zero.

Problem 9136

Problem 9137

Find the Tangent T⃗\vec{T}T, Normal N⃗\vec{N}N, and Binormal B⃗\vec{B}B vectors for r⃗(t)=⟨3cos⁡(2t),3sin⁡(2t),4t⟩\vec{r}(t)=\langle 3 \cos (2 t), 3 \sin (2 t), 4 t\rangler(t)=⟨3cos(2t),3sin(2t),4t⟩ at t=0t=0t=0.

Problem 9138

Find the Tangent, Normal, and Binormal vectors T⃗\vec{T}T, N⃗\vec{N}N, and B⃗\vec{B}B for r⃗(t)=⟨4cos⁡(2t),4sin⁡(2t),4t⟩\vec{r}(t)=\langle 4 \cos (2 t), 4 \sin (2 t), 4 t\rangler(t)=⟨4cos(2t),4sin(2t),4t⟩ at t=0t=0t=0.

Problem 9139

Find the Tangent, Normal, and Binormal vectors T⃗\vec{T}T, N⃗\vec{N}N, and B⃗\vec{B}B for r⃗(t)=⟨3cos⁡(2t),3sin⁡(2t),4t⟩\vec{r}(t)=\langle 3 \cos (2 t), 3 \sin (2 t), 4 t\rangler(t)=⟨3cos(2t),3sin(2t),4t⟩ at t=0t=0t=0.

Problem 9140

Find the curvature of the curve r⃗(t)=⟨cos⁡(2t),sin⁡(2t),t⟩\vec{r}(t)=\langle\cos (2 t), \sin (2 t), t\rangler(t)=⟨cos(2t),sin(2t),t⟩ at t=0t=0t=0. Round to two decimal places.

Problem 9141

Evaluate the integral: I=12∫sin⁡(2θ)sin⁡(θ)dθI = \frac{1}{2} \int \sin(2\theta) \sin(\theta) d\thetaI=21​∫sin(2θ)sin(θ)dθ.

Find the arc length $s$ of the curve $\vec{r}(t)=\langle 5 \sin (5 t), 5 \cos (5 t), 3 t\rangle$ for $-1 \leq t \leq 1$ .

Find the arc length $s$ of the curve $\vec{r}(t)=\langle-4 t+5,-4 t-2,-3 t-4\rangle$ for $-1 \leq t \leq 1$ .

Find the local extrema of $g(x, y)=x^{3}+22.5 x^{2}-1.5 y^{2}+150 x-18 y+70$ by calculating its partial derivatives.

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle-5 e^{3 t},-e^{3 t}, 2 e^{3 t}\rangle$ for $0 \leq t \leq \ln(4)$ . $s=$

Calculate the arclength of the curve $\vec{r}(t)=\left\langle 6 t, \frac{t^{3}}{3}, \frac{\sqrt{12} t^{2}}{2}\right\rangle$ for $-1 \leq t \leq 4$ .

Calculate the arclength of the curve $\vec{r}(t)=\left\langle 6 t, \frac{t^{3}}{3}, \frac{\sqrt{12} t^{2}}{2}\right\rangle$ for $-1 \leq t \leq 4$ .

Differentiate $y=\left(x^{2}+2\right)^{2}\left(x^{4}+4\right)^{4}$ using logarithmic differentiation.

Calculate the arclength of $\vec{r}(t)=\left\langle 3 t, \frac{t^{3}}{3}, \frac{\sqrt{6} t^{2}}{2}\right\rangle$ for $-3 \leq t \leq 1$ .

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle 12 \cos (-t), 2 \sqrt{13} \cos (-t), 14 \sin (-t)\rangle$ from $t=0$ to $t=\pi$ . $s=$

Find the derivative of $y=\frac{e^{-x} \cos ^{2} x}{x^{2}+x+1}$ using logarithmic differentiation.

Differentiate $y=x^{\sin x}$ using logarithmic differentiation.

Find the arc length function $s(t)$ and the arc length parameterization $\vec{r}(s)$ for the curve $\vec{r}(t)=\langle\cos (t), \sin (t), t\rangle$ .

Calculate the arclength of the curve $\vec{r}(t)=\left\langle 7 t, \frac{t^{3}}{3}, \frac{\sqrt{14} t^{2}}{2}\right\rangle$ for $t$ in $[-1, 4]$ .

Prove that $\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}$ .

Find the arc length function $s(t)$ and the arc length parameterization $\vec{r}(s)$ for the curve $\vec{r}(t)=\langle\cos (t), \sin (t), t\rangle$ .

Bestimmen Sie die Tangentengleichung an $K$ für $f(x)=x^{2}-3x+2$ im Punkt P $(4 \mid f(4))$ .

Find the arc length function $s(t)$ for the curve $\vec{r}(t)=\langle\cos(t), \sin(t), t\rangle$ and its arc length parameterization.

Find the derivative of $f(x)=\frac{x^{5}-x}{x^{3}}$ .

Calculate the arclength of the curve $\vec{r}(t)=\left\langle 6 t, \frac{t^{3}}{3}, \frac{\sqrt{12} t^{2}}{2}\right\rangle$ for $-2 \leq t \leq 3$ .

Find the derivative of $f(x)=x \cdot \sin(x)$ using the product rule.

Bestimme die Tangentengleichung an $K$ von $f$ im Punkt $A(-2 \mid \ldots)$ für $f(x)=\frac{1}{4} x^{4}-\frac{3}{2} x^{2}$ .

A fish is reeled in at 3.7 ft/s from 10 ft above water. Find the rate of angle change when 25 ft of line is out. $\frac{d \theta}{d t}=$ $\mathrm{rad} / \mathrm{sec}$

Find the following for the function $f(x)=5 x^{2}-9$ : a. $\frac{f(x)-f(a)}{x-a}$ ; b. $\frac{f(x+h)-f(x)}{h}$ .

Find the curvature of the curve $\vec{r}(t)=\langle t-\sin t, 1-\cos t\rangle$ at $t=\frac{7 \pi}{4}$ .

Find the curvature of the curve $\vec{r}(t)=\langle 3t, -4-3t, t \rangle$ at $t=-5$ .

Find the arc length function $s(t)$ and the arc length parameterization $\vec{r}(s)$ for the curve $\vec{r}(t)=\langle 5 \cos (t), 7 \sin (t), 2 \sqrt{6} \cos (t)\rangle$ .

Find the curvature of the curve $\vec{r}(t)=\langle 4 \cos (3 t), 4 \sin (3 t), t\rangle$ at $t=0$ , rounded to two decimal places.

Find the arc length function $s(t)$ and the arc length parameterization $\vec{r}(s)$ for the curve $\vec{r}(t)=\langle-4 t+3,3 t-3,-5 t+1\rangle$ .

Find the curvature of the curve $\vec{r}(t)=\langle -t, 4t^4, -3t^2 \rangle$ at $t=-2$ . Round to 2 decimal places.

Find the following for the function $f(x)=4x+8$ : a. $\frac{f(x)-f(a)}{x-a}$ , b. $\frac{f(x+h)-f(x)}{h}$ .

Determine which function has a horizontal asymptote at $y=0$ : $f(x)=\frac{x^{2}+2}{x-1}$ or $g(x)=e^{x}$ . Choose (A), (B), (C), or (D).

Calculate the curvature of the curve $\vec{r}(t)=\langle 1-3 \cos t, 0,4+3 \sin t\rangle$ at $t=\frac{1}{4} \pi$ .

Find the leading term in the Taylor series of $f(x)=\int_{0}^{x} \ln (\cosh (t)) dt$ around zero.

Find the Tangent $\vec{T}$ , Normal $\vec{N}$ , and Binormal $\vec{B}$ vectors for $\vec{r}(t)=\langle 3 \cos (2 t), 3 \sin (2 t), 4 t\rangle$ at $t=0$ .

Find the Tangent, Normal, and Binormal vectors $\vec{T}$ , $\vec{N}$ , and $\vec{B}$ for $\vec{r}(t)=\langle 4 \cos (2 t), 4 \sin (2 t), 4 t\rangle$ at $t=0$ .

Find the Tangent, Normal, and Binormal vectors $\vec{T}$ , $\vec{N}$ , and $\vec{B}$ for $\vec{r}(t)=\langle 3 \cos (2 t), 3 \sin (2 t), 4 t\rangle$ at $t=0$ .

Find the curvature of the curve $\vec{r}(t)=\langle\cos (2 t), \sin (2 t), t\rangle$ at $t=0$ . Round to two decimal places.

Evaluate the integral: $I = \frac{1}{2} \int \sin(2\theta) \sin(\theta) d\theta$ .

Find the velocity and acceleration vectors for $\vec{r}(t)=\langle-2 t, 3 t^{4}, 5 t^{3}+5\rangle$ at $t=-1$ .

Determine the acceleration vector for $\vec{r}(t)=\left\langle\sin (6 t), 11 t^{5}, e^{-3 t}\right\rangle$ . Find each component.

Find the velocity vector for $\vec{r}(t)=\langle\sin(8t), 10t^{10}, e^{-4t}\rangle$ . Compute each component.

Find the speed of the object with position $\bar{r}(t)=\langle 7 t^{2}+7,4 t-6,t^{3}+7 t\rangle$ at $t=4$ . Show answer to 4 decimal places. Speed =

Find the tangent line equation for the function $f(x)=\frac{x-13}{5x+6}$ at $x=3$ . Tangent line: $y=$

Gegeben ist die Funktion $f$ mit $f(x)=x^{2}-2$ .
a) Finde die Tangentengleichung an $P(0,5 \mid f(0,5)$ ). b) Bestimme Punkte mit Steigung 4 und 0. c) Wo ist die Tangente parallel zu $y=-2x+3$ ?

Approximate $\sqrt{25.3}$ using the linear approximation of $f(x)=\sqrt{x}$ at $x=25$ . Find $m$ and $b$ for $L(x)=m x+b$ .

Find the horizontal velocity, vertical velocity, and speed of an object at $t=2$ given $x=4t^{2}+3t$ , $y=5t^{2}+1$ .

Find the derivative $Y'(u)$ of the function $Y(u)=(u^{-2}+u^{-3})(u^{5}+5u^{2})$ .

Find points on the curve where the tangent line is parallel to $y=-2x$ for the equation $2x+y+x \frac{dy}{dx}+2y \frac{dy}{dx}=0$ .

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x}$

Find the degree 3 MacLaurin polynomial for $f(x)=e^{2x}$ and use it to approximate $e^{6}$ . $T_{3}(x)=\square+\square x+\square x^{2}+\square x^{3}$ $e^{6} \approx$

Find the area between $f(x)=e^{x} \sec ^{2} x$ and $g(x)=e^{x} \tan ^{2} x$ from $0$ to $\pi$ , avoiding singularities.

Find the tangential and normal components of the acceleration vector for $\vec{r}(t)=\langle-3t, t^5, t^2\rangle$ at $t=2$ .

Given $f(1)=2$ and $f^{\prime}(1)=3$ , find $g^{\prime}(1)$ and $h^{\prime}(1)$ , then the tangent lines for $g(x)$ and $h(x)$ .

A baseball is hit from 4 feet high at 116 ft/s and $34^{\circ}$ . What is its max height? Neglect air resistance.

Gegeben ist $Q(0)=1 \mathrm{As}$ .
a) Finde $Q(t)$ allgemein.
b) Berechne $Q(t)$ für $I_{0}=1 \mathrm{A}$ und $t_{1}, t_{2}, t_{3}$ .
c) Zeichne den Ladungsverlauf.

Gegeben ist ein Stromimpuls mit $Q(0)=Q_{0}=1$ As. Berechnen Sie $Q(t)$ allgemein und für $I_{0}=1 \mathrm{~A}$ , $t_{1}=1 \mathrm{~s}$ , $t_{2}=2 \mathrm{~s}$ , $t_{3}=3 \mathrm{~s}$ . Stellen Sie $Q(t)$ grafisch dar.

Find the second derivative $y^{\prime \prime}$ for the function $y=\sqrt{9 x+10}$ . Show your work.

Find the derivatives of the following functions with constant $a$ : (a) $y=x^{a}$ , (b) $y=a^{x}$ , (c) $y=x^{x}$ , (d) $y=a^{a}$ .

Find the derivative of $y$ with respect to $x$ for the function $y=4^{x}$ .

Differentiate the function: $\ln \left(3 x^{5} e^{2}\right)$ .

Evaluate the infinite series: $\sum_{n=1}^{\infty} \frac{1+3^{n}}{3^{n}+2^{n}}$ .

Find the derivatives and simplify: (a) $\frac{d}{d x}\left[e^{2 \ln x}\right]$ (b) $\frac{d}{d x}\left[\log _{a} a^{\sin x}\right]$ (c) $\frac{d}{d x}\left[\log _{2} 8^{x-5}\right]$

At noon, ship A is 180 km west of ship B. A sails south at 20 km/h, B sails north at 40 km/h. What's the distance change rate at 4 PM? $\square$ km/h

Find the derivative of $y$ for $y=7^{\sqrt{t}}$ with respect to $t$ .

What is the speed of a ball dropped from a height of $15 \mathrm{~m}$ when it hits the ground?

Find $dy/dx$ using implicit differentiation for the equation $2xy - y^2 = 1$ . Show your work.

Find the derivative of the function $f(x)=x^{2} \cdot \arctan (5 x)$ .