Calculus

Problem 6001

Ergänzen Sie die Tabelle mit den Differenzenquotienten und Limes für die Funktionen $f(x)=5x^{2}$ , $f(x)=4x^{2}-6$ und $f(x)=\frac{((1+h)-1)^{3}-(1-1)^{3}}{h}$ . Zusatz: Umformen, sodass $h$ nicht im Nenner steht.

See Solution

Problem 6002

Berechne die 1. Ableitung von $f$ an $x_0$ als Grenzwert des Differenzenquotienten für: a) $f(x)=0,25 x^{2}$ , $x_{0}=2$ b) $f(x)=0,5 x^{2}-1$ , $x_{0}=-1$

See Solution

Problem 6003

Find the rate of change of $f(t)=\frac{94 t}{99 t-94}$ after 1 hour. Answer in facts/hour, rounded to the nearest integer.

See Solution

Problem 6004

Ein Fallschirmspringer springt aus 4000 m. Berechne nach 10 s die Fallhöhe, die Höhe und die Zeit bis 1000 m erreicht werden.

See Solution

Problem 6005

Find $\lim _{x \rightarrow \infty} \frac{\sin \left(1+x^{4}\right)}{1+x^{4}}$ and explain your reasoning.

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Problem 6006

Ein Fallschirmspringer springt aus 4000 m Höhe. Berechne nach 10 s die gefallene Strecke, Höhe und die Zeit bis 1000 m.

See Solution

Problem 6007

Ein Fallschirmspringer springt aus $4000 \mathrm{~m}$ . a) Strecke und Höhe nach 10s? b) Höhe $h(t)$ ? c) Zeit bis $1000 \mathrm{~m}$ ?

See Solution

Problem 6008

A steel sheet (3.6 mm thick) has nitrogen at 120°C. Given diffusion coefficient $6.1 \times 10^{-11} \mathrm{~m}^{2}/\mathrm{s}$ , find nitrogen flux.

See Solution

Problem 6009

Continuity Quiz: Determine the nature of the discontinuity for $w(t)$ defined as $w(t)=\left\{\begin{array}{ll} t^{2}, & t<4 \\ 2 t, & t>4 \end{array}\right.$ at $t=4$ .

See Solution

Problem 6010

Which function is continuous on $3 \leq x \leq 7$ ? a. $y=\cos (x)$ b. $y=\ln (x-3)$ c. $y=\frac{\sqrt{x-3}}{x-7}$ d. $y=\frac{5}{x^{2}-9}$

See Solution

Problem 6011

Find the vertical asymptotes of $f(x)=\frac{2 x^{2}-6 x}{x^{2}-9}$ and justify using limits.

See Solution

Problem 6012

Find the average rate of change of $f(x)=3 x^{3}+2 x^{2}$ on the interval $[1,3]$ . Choices: a. 2, b. 17, c. 5, d. 47.

See Solution

Problem 6013

Find $a$ such that the average rate of change of $f(x)=6x^2+7x-5$ from $0$ to $18$ equals the instantaneous rate at $x=a$ . Options: a. 6, b. 19, c. 9, d. 8.

See Solution

Problem 6014

Leiten Sie die Funktion $f$ ab: a) $f(x)=x \cdot \sin (3 x)$ b) $f(x)=(2 x-1)^{2} \cdot \sqrt{x}$ c) $f(x)=3 x^{5} \cdot \cos (2 x)$ d) $f(x)=3 x \cdot \sin (4 x-1)$ e) $f(x)=(4-3 x)^{2} \cdot \sin (x)$ f) $f(x)=0,5 x^{2} \cdot \sqrt{4-x}$ g) $f(x)=x^{2} \cdot \cos (1-x)$ h) $f(x)=\sqrt{2 x+3} \cdot x^{2}$ i) $f(x)=(5 x+2)^{7} \cdot \cos (x)$

See Solution

Problem 6015

Find points on the curve $y=\cot \left(\frac{x}{5}\right)$ where the tangent is parallel to $x+5y=18$ .

See Solution

Problem 6016

Find $\frac{d y}{d t}$ given $y^{2}-(x+1)^{2}=\sin (x y)$ , $\frac{d x}{d t}=-\frac{4}{5}$ , $x=0$ , $y=1$ .

See Solution

Problem 6017

Berechnen Sie das Integral: $\int_{-2}^{-1}\left(-2 x^{-3}+3 x^{-3}\right) d x$

See Solution

Problem 6018

Find the $50^{\text{th}}$ derivative of the function $f(x) = \cos(2x)$ .

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Problem 6019

Berechne das Integral: $\int_{1}^{e}\left(\frac{1}{2 x}+0,5 x^{-1}\right) d x$

See Solution

Problem 6020

Find the derivative of the function $f(x)=e^{-x}$ . What is $f^{\prime}(x)$ ?

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Problem 6021

A car's velocity increased from $8.00 \, m/s$ to $16.0 \, m/s$ in $5.40 \, s$ . Find the distance traveled in that time.

See Solution

Problem 6022

A pool is 10 ft wide and 15 ft long. Find $V$ in terms of $h$ and rates $\frac{dV}{dt}$ and $\frac{dh}{dt}$ . If $h$ rises at $\frac{1}{10} \frac{\mathrm{ft}}{\min}$ , find $\frac{dV}{dt}$ . If filled at $20 \frac{\mathrm{ft}^3}{\mathrm{min}}$ , find $\frac{dh}{dt}$ .

See Solution

Problem 6023

Calculate the limit: $\lim _{t \rightarrow 0}\left(\frac{3}{t \sqrt{1+t}}-\frac{3}{t}\right)$ .

See Solution

Problem 6024

Evaluate the limit: $\lim _{t \rightarrow 0}\left(\frac{3}{t \sqrt{1+t}}-\frac{3}{t}\right)$ .

See Solution

Problem 6025

Evaluate the limit:
$\lim _{t \rightarrow 0}\left(\frac{3}{t \sqrt{1+t}}-\frac{3}{t}\right)$
What is the result?

See Solution

Problem 6026

Graph the function $g(x)$ where $\lim _{x \rightarrow 2^{-}} g(x)=-\infty$ .

See Solution

Problem 6027

Find the average rate of change of $f(x)=\frac{(3 x-2)(x+5)}{x^{2}+6 x+5}$ from $x=-6$ to $x=-3$ . Is it 0.5?

See Solution

Problem 6028

A spacecraft with mass $2.8 \times 10^{4} \mathrm{~kg}$ moves at $<0,26,0>\mathrm{km/s}$ . After firing thrusters for $21.5 \mathrm{~s}$ with force $<8 \times 10^{5}, 0,0>\mathrm{N}$ , where will it be in one hour?

See Solution

Problem 6029

At a clothing company, the cost function is $C(x)=\frac{x^{2}-7x+30}{x}$ .
a) Find the cost of 3000 jeans. b) Calculate the cost change rate at 3000 jeans.

See Solution

Problem 6030

Taylor Expand $100 + 3x + 8x^2 + 3x^3 + 4x^4$ at $x = -2$ .

See Solution

Problem 6031

Expand $4 x^{4}+3 x^{3}+8 x^{2}+3 x+100$ using Taylor series at $x=-2$ .

See Solution

Problem 6032

Find the instantaneous rate of change of $f(x)$ at $x=1$ using approximations. Justify your result. $f(x)=\frac{1}{5} \cos 2 x+4$

See Solution

Problem 6033

Find the derivative of the function $f(x) = 2 e^{k - x^{2}}$ .

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Problem 6034

Find the function $f$ if $f'''(x) = 2$ and it has a saddle point at $S(2, -\frac{1}{3})$ .

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Problem 6035

Find the second derivative of $g(t)=(t^{3}+1)^{2}$ . What is $g^{\prime \prime}(t)$ ? Options: A. $4 t^{2}$ B. $12 t (t^{3}+1)+18 t^{4}$ C. $6 t^{2} (t^{3}+1)+4 t^{2}$ D. $4 t^{2}+1$ E. None.

See Solution

Problem 6036

Person A is 100 feet west of a runner moving north at 9 ft/s. After 10 seconds, find the rate of distance change in ft/s.

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Problem 6037

A balloon leaks at $50 \mathrm{~cm}^{3} / \mathrm{s}$ . Find the radius change rate (in $\mathrm{cm} / \mathrm{s}$ ) when radius is $10 \mathrm{~cm}$ . Round to 0.000 place.

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Problem 6038

Water flows from a cone funnel (4m height, 8m radius) at $0.8 \mathrm{~m}^{3} / \mathrm{min}$ . Find the radius change rate when water height is $3 \mathrm{~m}$ . Answer to 0.0000 decimal places.

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Problem 6039

Find the derivative of $y=3 \cos x \cot x$ . What is $\frac{d y}{d x}$ ?

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Problem 6040

Find the derivative of the function $y=5 \sec x+3 \csc x$ .

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Problem 6041

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{\cos x}{1+\cos x}$ .

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Problem 6042

Find the derivative of $f(x)=6 \tan(7-x^{2})$ with respect to $x$ .

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Problem 6043

Find the derivative of $f(x)=\cot (3-9 x)$ with respect to $x$ .

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Problem 6044

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-5 x+1}-x\right)=$

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Problem 6045

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{6}{e^{x}+3}=$

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Problem 6046

Calculate the limit: $\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}-5 x+1}-x\right)$ .

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Problem 6047

What would \ $29 invested at 3\% continuous interest in 1623 be worth in 2000? Use the formula$ A = Pe^{rt}$.

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Problem 6048

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{6}{e^{x}+3}$ and (b) $\lim _{x \rightarrow-\infty} \frac{6}{e^{x}+3}$ .

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Problem 6049

Find the derivative of $h(s)=\sin ^{8} s+\cos ^{7} s$ with respect to $s$ .

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Problem 6050

Find the derivative of $f(x)=\frac{\cos x^{3}}{\cos ^{9} x}$ with respect to $x$ .

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Problem 6051

Find the derivative of $f(x)=-8 \csc (3-6 x)$ with respect to $x$ .

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Problem 6052

Find the limit: $\lim _{x \rightarrow 0^{+}}\left(\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x^{2}+x}}\right)$ .

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Problem 6053

Find the derivative of $f(x)=\frac{\cos x^{8}}{\cos ^{7} x}$ with respect to $x$ .

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Problem 6054

Investing $\mathrm{P}_{0}$ at $6.9\%$ interest, find $P(t)$ , balance after 1 year for \$1500, and when it doubles.

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Problem 6055

Find the derivative of $f(x)=\cos^{2}(6x^{2}-4)$ with respect to $x$ .

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Problem 6056

Find the derivative $y^{\prime}$ for the function $y=\sec^{5}(5x)$ .

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Problem 6057

Find the derivative of the function $f(x)=\sqrt{7+x \sin x}$ .

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Problem 6058

Find the derivative $\frac{d y}{d t}$ for $y=\sin (\tan (5 t-6))$ .

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Problem 6059

Find the derivative of the function $f(x)=9 \sin \sqrt{x^{2}+9}$ .

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Problem 6060

Find the derivative $d$ of the function $y=\frac{e^{7u}-e^{-7u}}{e^{7u}+e^{-7u}}$ .

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Problem 6061

Determine the discontinuity of the piecewise function $g(x)=\left\{\begin{array}{ll}x^{3} & \text { if } x<0 \\ e^{x} & \text { if } x>0\end{array}\right.$ .

See Solution

Problem 6062

Find the sum of the series: $\sum_{i=0}^{\infty}\left(\frac{1}{2^{i}}\right)$ . Choose (a) 1, (b) $\frac{1}{2}$ , (c) $\frac{1}{4}$ , (d) 2.

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Problem 6063

Find the sum of the series $1+e^{-1}+e^{-2}+e^{-3}+\ldots$ from the options: (a) $1+\frac{1}{c}$ (b) $\frac{e}{e+1}$ (c) $1-\frac{1}{e}$ (d) $\frac{e}{e-1}$ .

See Solution

Problem 6064

Find the half-life of a substance with a decay rate of $8.9\%$ per day using the continuous exponential decay model.

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Problem 6065

Find the half-life of a radioactive substance with a decay rate of $8.9\%$ per day.

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Problem 6066

Find $\frac{d z}{d t}$ given $x^{2}+y^{2}+z^{2}=9$ , $\frac{d x}{d t}=4$ , $\frac{d y}{d t}=3$ , at $(2,2,1)$ .

See Solution

Problem 6067

A rectangle's length increases by $9 \mathrm{~cm/s}$ and width by $5 \mathrm{~cm/s}$ . Find area growth rate when length is $7 \mathrm{~cm}$ and width is $4 \mathrm{~cm}$ .

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Problem 6068

Set up a definite integral using the disk/washer method for the volume of the solid formed by rotating the region bounded by $x=2\sqrt{y}$ , $x=0$ , $y=9$ , and the line $x=7$ .

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Problem 6069

Find the centripetal acceleration in terms of $\mathrm{g}$ for a stone rotated every $0.44 \mathrm{s}$ on a $1.14 \mathrm{m}$ rope.

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Problem 6070

Use Boyle's Law $PV=C$ . Given $V=300 \mathrm{~cm}^{3}$ , $P=100 \mathrm{kPa}$ , and $\frac{dP}{dt}=10 \mathrm{kPa/min}$ , find $\frac{dV}{dt}$ .

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Problem 6071

Find centripetal acceleration as a multiple of $g$ for a plane at $1727 \mathrm{~km/h}$ on a curve with radius $8 \mathrm{~km}$ .

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Problem 6072

Find the volume of the solid formed by rotating the area between $y=x^{2}$ and $y=2x$ about various lines.

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Problem 6073

Find the volume of the solid with a base under $y=1-x^{2}$ and the x-axis, with square cross sections perpendicular to x-axis.

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Problem 6074

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(1+\frac{4}{x}\right)^{\frac{x}{3}}$ . Use L'Hospital's rule if needed.

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Problem 6075

Evaluate the integral: $\int (1 - 2x^{2} + x^{4}) \, dx$

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Problem 6076

Evaluate the expression $x - \frac{2}{3} x^{3} + \frac{x^{5}}{5}$ from $x = -1$ to $x = 1$ .

See Solution

Problem 6077

Find the volume of the solid with a base under $y=1-x^{2}$ and the $x$ -axis, with square cross sections.

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Problem 6078

若 $f(x)$ 是 $g(x)$ 的原函数，哪项正确？A. $\int f(x) d x=g(x)+c$ B. $\int g(x) d x=f(x)+c$ C. $\int f(x) d x=g(x)$ D. $\int g(x) d x=f(x)$

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Problem 6079

Find the discontinuity point of the piecewise function: $f(x) = \begin{cases} \cos x & \text{if } x \leq 0 \\ e^{x} & \text{if } x > 0 \end{cases}$ .

See Solution

Problem 6080

Find the derivative of the equation $9xy + 7x^3 + 2y^2 = 8$ with respect to $x$ .

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Problem 6081

Find the constant value of $\frac{\frac{d^{2} y}{d x^{2}}}{\left(\frac{d y}{d x}\right)^{4}}$ for the curve defined by $x=t^{3}+1$ and $y=t^{2}+1$ .

See Solution

Problem 6082

Calculate the partial derivative of $9xy + 7x^3 + 2y^2 = 8$ with respect to $y$ .

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Problem 6083

In a snowball fight, if you throw one snowball at $61^{\circ}$ with speed $11 \mathrm{~m/s}$ , what angle should the second snowball be thrown at? Also, how many seconds after the first should it be thrown? Use $g = 9.8 \mathrm{~m/s^2}$ .

See Solution

Problem 6084

Set up a definite integral for the volume of the solid formed by rotating the area between $x=2\sqrt{y}$ , $x=0$ , $y=9$ around $x=7$ .

See Solution

Problem 6085

Set up a definite integral using the disk/washer method for the volume of the solid formed by rotating the region bounded by $x=2\sqrt{y}$ , $x=0$ , $y=9$ , and the line $y=-3$ .

See Solution

Problem 6086

Find the average rate of change of $f(x)=7 x^{2}-5$ from $x=4$ to $x=a$ . Express your answer in terms of $a$ .

See Solution

Problem 6087

Between 2006-2016, patent applications $N$ grew by $3.9\%$ yearly.
a) Find $N(t)=444,000 e^{0.039 t}$ . b) Estimate $N$ in 2020. c) Estimate the growth rate in 2020.

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Problem 6088

已知函数 $f(x)=\left\{\begin{array}{l}\frac{1}{3} x^{3}+m x^{2}(x \leqslant 0,0 \leqslant m<1), \\ \mathrm{e}^{x-1}(x>0) .\end{array}\right.$ 求切线方程和极值及区间 $[-2,0]$ 上的最值.

See Solution

Problem 6089

Find the one-sided limit: $\lim _{x \rightarrow(1 / 2)^{+}} 15 x^{2} \tan \pi x$ .

See Solution

Problem 6090

Prove that $\lim _{x \rightarrow 5^{-}} \frac{1}{x-5}=-\infty$ using the $\varepsilon-\delta$ definition of limits.

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Problem 6091

A 10m ladder leans against a house. If its base moves away at 0.5m/s, find rates $r$ for $x=6, 8$ and limit as $x$ approaches 10.
$r=\frac{0.5 x}{\sqrt{100-x^{2}}} \mathrm{~m/s}$

See Solution

Problem 6092

Estimate the slope between the points $\left(-\frac{1}{2}, \frac{1}{3}\right)$ and $\left(0.90, 0\right)$ .

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Problem 6093

Calculate the limit: $\lim _{x \rightarrow 1}\left(\frac{1}{x-1}-\frac{1}{x+1}\right)$ .

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Problem 6094

Find the derivative of $f(x)=x^{3}+5 x^{2}+6$ at $x=-5$ using the alternative derivative form.

See Solution

Problem 6095

Find the derivative of $f(x) = x^{3} + 5x^{2} + 6$ at $x = c = -5$ using the alternative form of the derivative.

See Solution

Problem 6096

Find the function $f(x)$ and the number $c$ given that the limit represents $f^{\prime}(c)$ :
$f(x)=\lim _{\Delta x \rightarrow 0} \frac{[3-2(1+\Delta x)]-1}{\Delta x}$

See Solution

Problem 6097

Find the tangent lines to $f(x)=10x - x^2$ through $(5,26)$ , one with positive slope and one with negative slope.

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Problem 6098

Find the derivative of $f(x)=\frac{1}{2} e^{\frac{1}{3} x+2}$ and simplify it.

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Problem 6099

Find the $x$ -values where the function $f(x)=(x+2)^{2 / 3}$ is differentiable. Use interval notation for your answer.

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Problem 6100

Find the derivative of $f(x)=\sqrt{7x-5}$ and simplify your answer.

See Solution

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Calculus

Problem 6001

Problem 6002

Berechne die 1. Ableitung von fff an x0x_0x0​ als Grenzwert des Differenzenquotienten für: a) f(x)=0,25x2f(x)=0,25 x^{2}f(x)=0,25x2, x0=2x_{0}=2x0​=2 b) f(x)=0,5x2−1f(x)=0,5 x^{2}-1f(x)=0,5x2−1, x0=−1x_{0}=-1x0​=−1

Problem 6003

Find the rate of change of f(t)=94t99t−94f(t)=\frac{94 t}{99 t-94}f(t)=99t−9494t​ after 1 hour. Answer in facts/hour, rounded to the nearest integer.

Problem 6004

Ein Fallschirmspringer springt aus 4000 m. Berechne nach 10 s die Fallhöhe, die Höhe und die Zeit bis 1000 m erreicht werden.

Problem 6005

Find lim⁡x→∞sin⁡(1+x4)1+x4\lim _{x \rightarrow \infty} \frac{\sin \left(1+x^{4}\right)}{1+x^{4}}limx→∞​1+x4sin(1+x4)​ and explain your reasoning.

Problem 6006

Ein Fallschirmspringer springt aus 4000 m Höhe. Berechne nach 10 s die gefallene Strecke, Höhe und die Zeit bis 1000 m.

Problem 6007

Ein Fallschirmspringer springt aus 4000 m4000 \mathrm{~m}4000 m. a) Strecke und Höhe nach 10s? b) Höhe h(t)h(t)h(t)? c) Zeit bis 1000 m1000 \mathrm{~m}1000 m?

Problem 6008

A steel sheet (3.6 mm thick) has nitrogen at 120°C. Given diffusion coefficient 6.1×10−11 m2/s6.1 \times 10^{-11} \mathrm{~m}^{2}/\mathrm{s}6.1×10−11 m2/s, find nitrogen flux.

Problem 6009

Continuity Quiz: Determine the nature of the discontinuity for w(t)w(t)w(t) defined as w(t)={t2,t<42t,t>4w(t)=\left\{\begin{array}{ll} t^{2}, & t<4 \\ 2 t, & t>4 \end{array}\right.w(t)={t2,2t,​t<4t>4​ at t=4t=4t=4.

Problem 6010

Which function is continuous on 3≤x≤73 \leq x \leq 73≤x≤7? a. y=cos⁡(x)y=\cos (x)y=cos(x) b. y=ln⁡(x−3)y=\ln (x-3)y=ln(x−3) c. y=x−3x−7y=\frac{\sqrt{x-3}}{x-7}y=x−7x−3​​ d. y=5x2−9y=\frac{5}{x^{2}-9}y=x2−95​

Problem 6011

Find the vertical asymptotes of f(x)=2x2−6xx2−9f(x)=\frac{2 x^{2}-6 x}{x^{2}-9}f(x)=x2−92x2−6x​ and justify using limits.

Problem 6012

Find the average rate of change of f(x)=3x3+2x2f(x)=3 x^{3}+2 x^{2}f(x)=3x3+2x2 on the interval [1,3][1,3][1,3]. Choices: a. 2, b. 17, c. 5, d. 47.

Problem 6013

Find aaa such that the average rate of change of f(x)=6x2+7x−5f(x)=6x^2+7x-5f(x)=6x2+7x−5 from 000 to 181818 equals the instantaneous rate at x=ax=ax=a. Options: a. 6, b. 19, c. 9, d. 8.

Problem 6014

Problem 6015

Find points on the curve y=cot⁡(x5)y=\cot \left(\frac{x}{5}\right)y=cot(5x​) where the tangent is parallel to x+5y=18x+5y=18x+5y=18.

Problem 6016

Find dydt\frac{d y}{d t}dtdy​ given y2−(x+1)2=sin⁡(xy)y^{2}-(x+1)^{2}=\sin (x y)y2−(x+1)2=sin(xy), dxdt=−45\frac{d x}{d t}=-\frac{4}{5}dtdx​=−54​, x=0x=0x=0, y=1y=1y=1.

Problem 6017

Berechnen Sie das Integral: ∫−2−1(−2x−3+3x−3)dx\int_{-2}^{-1}\left(-2 x^{-3}+3 x^{-3}\right) d x∫−2−1​(−2x−3+3x−3)dx

Problem 6018

Find the 50th50^{\text{th}}50th derivative of the function f(x)=cos⁡(2x)f(x) = \cos(2x)f(x)=cos(2x).

Problem 6019

Berechne das Integral: ∫1e(12x+0,5x−1)dx\int_{1}^{e}\left(\frac{1}{2 x}+0,5 x^{-1}\right) d x∫1e​(2x1​+0,5x−1)dx

Problem 6020

Find the derivative of the function f(x)=e−xf(x)=e^{-x}f(x)=e−x. What is f′(x)f^{\prime}(x)f′(x)?

Problem 6021

A car's velocity increased from 8.00 m/s8.00 \, m/s8.00m/s to 16.0 m/s16.0 \, m/s16.0m/s in 5.40 s5.40 \, s5.40s. Find the distance traveled in that time.

Problem 6022

Problem 6023

Calculate the limit: lim⁡t→0(3t1+t−3t)\lim _{t \rightarrow 0}\left(\frac{3}{t \sqrt{1+t}}-\frac{3}{t}\right)limt→0​(t1+t​3​−t3​).

Problem 6024

Evaluate the limit: lim⁡t→0(3t1+t−3t)\lim _{t \rightarrow 0}\left(\frac{3}{t \sqrt{1+t}}-\frac{3}{t}\right)t→0lim​(t1+t​3​−t3​).

Problem 6025

Evaluate the limit: lim⁡t→0(3t1+t−3t) \lim _{t \rightarrow 0}\left(\frac{3}{t \sqrt{1+t}}-\frac{3}{t}\right) t→0lim​(t1+t​3​−t3​) What is the result?

Problem 6026

Graph the function g(x)g(x)g(x) where lim⁡x→2−g(x)=−∞\lim _{x \rightarrow 2^{-}} g(x)=-\inftylimx→2−​g(x)=−∞.

Problem 6027

Find the average rate of change of f(x)=(3x−2)(x+5)x2+6x+5f(x)=\frac{(3 x-2)(x+5)}{x^{2}+6 x+5}f(x)=x2+6x+5(3x−2)(x+5)​ from x=−6x=-6x=−6 to x=−3x=-3x=−3. Is it 0.5?

Problem 6028

A spacecraft with mass 2.8×104 kg2.8 \times 10^{4} \mathrm{~kg}2.8×104 kg moves at <0,26,0>km/s<0,26,0>\mathrm{km/s}<0,26,0>km/s. After firing thrusters for 21.5 s21.5 \mathrm{~s}21.5 s with force <8×105,0,0>N<8 \times 10^{5}, 0,0>\mathrm{N}<8×105,0,0>N, where will it be in one hour?

Problem 6029

At a clothing company, the cost function is C(x)=x2−7x+30xC(x)=\frac{x^{2}-7x+30}{x}C(x)=xx2−7x+30​. a) Find the cost of 3000 jeans. b) Calculate the cost change rate at 3000 jeans.

Problem 6030

Taylor Expand 100+3x+8x2+3x3+4x4100 + 3x + 8x^2 + 3x^3 + 4x^4100+3x+8x2+3x3+4x4 at x=−2x = -2x=−2.

Problem 6031

Expand 4x4+3x3+8x2+3x+1004 x^{4}+3 x^{3}+8 x^{2}+3 x+1004x4+3x3+8x2+3x+100 using Taylor series at x=−2x=-2x=−2.

Problem 6032

Find the instantaneous rate of change of f(x)f(x)f(x) at x=1x=1x=1 using approximations. Justify your result. f(x)=15cos⁡2x+4f(x)=\frac{1}{5} \cos 2 x+4f(x)=51​cos2x+4

Problem 6033

Find the derivative of the function f(x)=2ek−x2f(x) = 2 e^{k - x^{2}}f(x)=2ek−x2.

Problem 6034

Find the function fff if f′′′(x)=2f'''(x) = 2f′′′(x)=2 and it has a saddle point at S(2,−13)S(2, -\frac{1}{3})S(2,−31​).

Problem 6035

Find the second derivative of g(t)=(t3+1)2g(t)=(t^{3}+1)^{2}g(t)=(t3+1)2. What is g′′(t)g^{\prime \prime}(t)g′′(t)? Options: A. 4t24 t^{2}4t2 B. 12t(t3+1)+18t412 t (t^{3}+1)+18 t^{4}12t(t3+1)+18t4 C. 6t2(t3+1)+4t26 t^{2} (t^{3}+1)+4 t^{2}6t2(t3+1)+4t2 D. 4t2+14 t^{2}+14t2+1 E. None.

Problem 6036

Person A is 100 feet west of a runner moving north at 9 ft/s. After 10 seconds, find the rate of distance change in ft/s.

Problem 6037

A balloon leaks at 50 cm3/s50 \mathrm{~cm}^{3} / \mathrm{s}50 cm3/s. Find the radius change rate (in cm/s\mathrm{cm} / \mathrm{s}cm/s) when radius is 10 cm10 \mathrm{~cm}10 cm. Round to 0.000 place.

Problem 6038

Water flows from a cone funnel (4m height, 8m radius) at 0.8 m3/min0.8 \mathrm{~m}^{3} / \mathrm{min}0.8 m3/min. Find the radius change rate when water height is 3 m3 \mathrm{~m}3 m. Answer to 0.0000 decimal places.

Problem 6039

Find the derivative of y=3cos⁡xcot⁡xy=3 \cos x \cot xy=3cosxcotx. What is dydx\frac{d y}{d x}dxdy​?

Problem 6040

Find the derivative of the function y=5sec⁡x+3csc⁡xy=5 \sec x+3 \csc xy=5secx+3cscx.

Problem 6041

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=cos⁡x1+cos⁡xy=\frac{\cos x}{1+\cos x}y=1+cosxcosx​.

Berechne die 1. Ableitung von $f$ an $x_0$ als Grenzwert des Differenzenquotienten für: a) $f(x)=0,25 x^{2}$ , $x_{0}=2$ b) $f(x)=0,5 x^{2}-1$ , $x_{0}=-1$

Find the rate of change of $f(t)=\frac{94 t}{99 t-94}$ after 1 hour. Answer in facts/hour, rounded to the nearest integer.

Find $\lim _{x \rightarrow \infty} \frac{\sin \left(1+x^{4}\right)}{1+x^{4}}$ and explain your reasoning.

Ein Fallschirmspringer springt aus $4000 \mathrm{~m}$ . a) Strecke und Höhe nach 10s? b) Höhe $h(t)$ ? c) Zeit bis $1000 \mathrm{~m}$ ?

A steel sheet (3.6 mm thick) has nitrogen at 120°C. Given diffusion coefficient $6.1 \times 10^{-11} \mathrm{~m}^{2}/\mathrm{s}$ , find nitrogen flux.

Continuity Quiz: Determine the nature of the discontinuity for $w(t)$ defined as $w(t)=\left\{\begin{array}{ll} t^{2}, & t<4 \\ 2 t, & t>4 \end{array}\right.$ at $t=4$ .

Which function is continuous on $3 \leq x \leq 7$ ? a. $y=\cos (x)$ b. $y=\ln (x-3)$ c. $y=\frac{\sqrt{x-3}}{x-7}$ d. $y=\frac{5}{x^{2}-9}$

Find the vertical asymptotes of $f(x)=\frac{2 x^{2}-6 x}{x^{2}-9}$ and justify using limits.

Find the average rate of change of $f(x)=3 x^{3}+2 x^{2}$ on the interval $[1,3]$ . Choices: a. 2, b. 17, c. 5, d. 47.

Find $a$ such that the average rate of change of $f(x)=6x^2+7x-5$ from $0$ to $18$ equals the instantaneous rate at $x=a$ . Options: a. 6, b. 19, c. 9, d. 8.

Find points on the curve $y=\cot \left(\frac{x}{5}\right)$ where the tangent is parallel to $x+5y=18$ .

Find $\frac{d y}{d t}$ given $y^{2}-(x+1)^{2}=\sin (x y)$ , $\frac{d x}{d t}=-\frac{4}{5}$ , $x=0$ , $y=1$ .

Berechnen Sie das Integral: $\int_{-2}^{-1}\left(-2 x^{-3}+3 x^{-3}\right) d x$

Find the $50^{\text{th}}$ derivative of the function $f(x) = \cos(2x)$ .

Berechne das Integral: $\int_{1}^{e}\left(\frac{1}{2 x}+0,5 x^{-1}\right) d x$

Find the derivative of the function $f(x)=e^{-x}$ . What is $f^{\prime}(x)$ ?

A car's velocity increased from $8.00 \, m/s$ to $16.0 \, m/s$ in $5.40 \, s$ . Find the distance traveled in that time.

Calculate the limit: $\lim _{t \rightarrow 0}\left(\frac{3}{t \sqrt{1+t}}-\frac{3}{t}\right)$ .

Evaluate the limit: $\lim _{t \rightarrow 0}\left(\frac{3}{t \sqrt{1+t}}-\frac{3}{t}\right)$ .

Evaluate the limit:
$\lim _{t \rightarrow 0}\left(\frac{3}{t \sqrt{1+t}}-\frac{3}{t}\right)$
What is the result?

Graph the function $g(x)$ where $\lim _{x \rightarrow 2^{-}} g(x)=-\infty$ .

Find the average rate of change of $f(x)=\frac{(3 x-2)(x+5)}{x^{2}+6 x+5}$ from $x=-6$ to $x=-3$ . Is it 0.5?

A spacecraft with mass $2.8 \times 10^{4} \mathrm{~kg}$ moves at $<0,26,0>\mathrm{km/s}$ . After firing thrusters for $21.5 \mathrm{~s}$ with force $<8 \times 10^{5}, 0,0>\mathrm{N}$ , where will it be in one hour?

At a clothing company, the cost function is $C(x)=\frac{x^{2}-7x+30}{x}$ .
a) Find the cost of 3000 jeans. b) Calculate the cost change rate at 3000 jeans.

Taylor Expand $100 + 3x + 8x^2 + 3x^3 + 4x^4$ at $x = -2$ .

Expand $4 x^{4}+3 x^{3}+8 x^{2}+3 x+100$ using Taylor series at $x=-2$ .

Find the instantaneous rate of change of $f(x)$ at $x=1$ using approximations. Justify your result. $f(x)=\frac{1}{5} \cos 2 x+4$

Find the derivative of the function $f(x) = 2 e^{k - x^{2}}$ .

Find the function $f$ if $f'''(x) = 2$ and it has a saddle point at $S(2, -\frac{1}{3})$ .

Find the second derivative of $g(t)=(t^{3}+1)^{2}$ . What is $g^{\prime \prime}(t)$ ? Options: A. $4 t^{2}$ B. $12 t (t^{3}+1)+18 t^{4}$ C. $6 t^{2} (t^{3}+1)+4 t^{2}$ D. $4 t^{2}+1$ E. None.

A balloon leaks at $50 \mathrm{~cm}^{3} / \mathrm{s}$ . Find the radius change rate (in $\mathrm{cm} / \mathrm{s}$ ) when radius is $10 \mathrm{~cm}$ . Round to 0.000 place.

Water flows from a cone funnel (4m height, 8m radius) at $0.8 \mathrm{~m}^{3} / \mathrm{min}$ . Find the radius change rate when water height is $3 \mathrm{~m}$ . Answer to 0.0000 decimal places.

Find the derivative of $y=3 \cos x \cot x$ . What is $\frac{d y}{d x}$ ?

Find the derivative of the function $y=5 \sec x+3 \csc x$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{\cos x}{1+\cos x}$ .

Find the derivative of $f(x)=6 \tan(7-x^{2})$ with respect to $x$ .

Find the derivative of $f(x)=\cot (3-9 x)$ with respect to $x$ .

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-5 x+1}-x\right)=$

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{6}{e^{x}+3}=$

Calculate the limit: $\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}-5 x+1}-x\right)$ .

What would \ $29 invested at 3\% continuous interest in 1623 be worth in 2000? Use the formula$ A = Pe^{rt}$.

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{6}{e^{x}+3}$ and (b) $\lim _{x \rightarrow-\infty} \frac{6}{e^{x}+3}$ .

Find the derivative of $h(s)=\sin ^{8} s+\cos ^{7} s$ with respect to $s$ .

Find the derivative of $f(x)=\frac{\cos x^{3}}{\cos ^{9} x}$ with respect to $x$ .

Find the derivative of $f(x)=-8 \csc (3-6 x)$ with respect to $x$ .

Find the limit: $\lim _{x \rightarrow 0^{+}}\left(\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x^{2}+x}}\right)$ .

Find the derivative of $f(x)=\frac{\cos x^{8}}{\cos ^{7} x}$ with respect to $x$ .

Investing $\mathrm{P}_{0}$ at $6.9\%$ interest, find $P(t)$ , balance after 1 year for \$1500, and when it doubles.

Find the derivative of $f(x)=\cos^{2}(6x^{2}-4)$ with respect to $x$ .

Find the derivative $y^{\prime}$ for the function $y=\sec^{5}(5x)$ .

Find the derivative of the function $f(x)=\sqrt{7+x \sin x}$ .

Find the derivative $\frac{d y}{d t}$ for $y=\sin (\tan (5 t-6))$ .

Find the derivative of the function $f(x)=9 \sin \sqrt{x^{2}+9}$ .

Find the derivative $d$ of the function $y=\frac{e^{7u}-e^{-7u}}{e^{7u}+e^{-7u}}$ .

Determine the discontinuity of the piecewise function $g(x)=\left\{\begin{array}{ll}x^{3} & \text { if } x<0 \\ e^{x} & \text { if } x>0\end{array}\right.$ .

Find the sum of the series: $\sum_{i=0}^{\infty}\left(\frac{1}{2^{i}}\right)$ . Choose (a) 1, (b) $\frac{1}{2}$ , (c) $\frac{1}{4}$ , (d) 2.

Find the sum of the series $1+e^{-1}+e^{-2}+e^{-3}+\ldots$ from the options: (a) $1+\frac{1}{c}$ (b) $\frac{e}{e+1}$ (c) $1-\frac{1}{e}$ (d) $\frac{e}{e-1}$ .

Find the half-life of a substance with a decay rate of $8.9\%$ per day using the continuous exponential decay model.

Find the half-life of a radioactive substance with a decay rate of $8.9\%$ per day.

Find $\frac{d z}{d t}$ given $x^{2}+y^{2}+z^{2}=9$ , $\frac{d x}{d t}=4$ , $\frac{d y}{d t}=3$ , at $(2,2,1)$ .

A rectangle's length increases by $9 \mathrm{~cm/s}$ and width by $5 \mathrm{~cm/s}$ . Find area growth rate when length is $7 \mathrm{~cm}$ and width is $4 \mathrm{~cm}$ .

Set up a definite integral using the disk/washer method for the volume of the solid formed by rotating the region bounded by $x=2\sqrt{y}$ , $x=0$ , $y=9$ , and the line $x=7$ .

Find the centripetal acceleration in terms of $\mathrm{g}$ for a stone rotated every $0.44 \mathrm{s}$ on a $1.14 \mathrm{m}$ rope.

Use Boyle's Law $PV=C$ . Given $V=300 \mathrm{~cm}^{3}$ , $P=100 \mathrm{kPa}$ , and $\frac{dP}{dt}=10 \mathrm{kPa/min}$ , find $\frac{dV}{dt}$ .

Find centripetal acceleration as a multiple of $g$ for a plane at $1727 \mathrm{~km/h}$ on a curve with radius $8 \mathrm{~km}$ .

Find the volume of the solid formed by rotating the area between $y=x^{2}$ and $y=2x$ about various lines.

Find the volume of the solid with a base under $y=1-x^{2}$ and the x-axis, with square cross sections perpendicular to x-axis.

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(1+\frac{4}{x}\right)^{\frac{x}{3}}$ . Use L'Hospital's rule if needed.