Calculus

Problem 22001

Find where the function $f(x)=3 x^{2}+(x-4)^{2 / 3}+\frac{1}{x}$ is continuous.

See Solution

Problem 22002

Find the tangent line equation for $f(x)=\frac{25}{x^{2}+1}$ at $x=2$ .

See Solution

Problem 22003

Find where the function $f(x) = 3x^{2} + (x-4)^{\frac{2}{3}} + \frac{1}{x}$ is continuous.

See Solution

Problem 22004

Inflate a balloon at 16 in³/s. Find the radius increase rate when it’s 3 inches. Give exact or rounded answer with units.

See Solution

Problem 22005

Evaluate the integral from 0 to 1 of $(2+x^{2})$ with respect to $x$ .

See Solution

Problem 22006

A rock thrown up on the moon at 32 m/s has height $s(t)=32t-0.8t^2$ . Solve for velocity, time to max height, and max height.

See Solution

Problem 22007

Find the drug concentration $\mathrm{C}(9)$ after 9 minutes using $C(t)=0.07(1-e^{-0.2 t})$ . Round to three decimals.

See Solution

Problem 22008

Find the absolute max and min of $f(x)=x^{3}+6x^{2}-2$ on the interval $-5 \leq x \leq 1$ .

See Solution

Problem 22009

Find the absolute max and min of $f(x) = x^{3} + 6x^{2} - 2$ on $[-5, 1]$ . Min = -30; Max = 2.

See Solution

Problem 22010

Find the elasticity of demand for $D(p) = 155 - p^{2}$ at $p = 11$ and classify it as elastic, inelastic, or unit elastic.

See Solution

Problem 22011

Find the production level $x$ that minimizes the average cost given $c(x)=5x^{3}-35x^{2}+5,547x$ .

See Solution

Problem 22012

Find the derivative of $e^{4 x^{2}+3 x y+y^{2}}$ with respect to $x$ .

See Solution

Problem 22013

Let $(\Delta v)_{n}=\frac{1}{n(n+1)}$ .
(a) Expand $\frac{1}{n(n+1)}$ using partial fractions. (b) Find $v_{n}$ based on (a). (c) Verify your result in (b) by finding the forward difference. (d) Write the first six terms of $(\Delta v)_{n}$ from (a) without combining fractions. (e) Calculate $\sum_{n=1}^{N}(\Delta v)_{n}$ for $N=2,3,4$ . (f) Relate the result from (e) to continuous calculus concepts.

See Solution

Problem 22014

Find the partial derivative $\frac{\partial}{\partial y} x^{2} \ln \left(2 x y+3 x+y^{3}\right)$ .

See Solution

Problem 22015

Find the derivative $\frac{\partial}{\partial x} x^{2} \ln (x y + 3 x + y^{2})$ .

See Solution

Problem 22016

Find the arc length of the curve $y=\frac{x^{6}}{6}+\frac{1}{16 x^{4}}$ on $[1,3]$ . Length: $\square$ .

See Solution

Problem 22017

112 rabbits are released, growing to 1008 in 2 years. What will the population be after another 9 months? Round to nearest rabbit.

See Solution

Problem 22018

Find the volume $V$ of the solid formed by rotating the area between $x=8y^2$ and $x=8$ around the line $x=8$ .

See Solution

Problem 22019

Find the arc length of the curve $y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}}$ on the interval $[1,4]$ .

See Solution

Problem 22020

Find the exponential growth rate of a bacteria culture that doubles in $8 \mathrm{hr}$ . Options: a) $5.5 \%$ b) $8.7 \%$ c) $3.8 \%$ d) $4.2 \%$

See Solution

Problem 22021

Find the tangent line equation to $y=f(x)$ at $x=e$ , where $f(x)=x-6 \ln x$ .

See Solution

Problem 22022

Evaluate the series $\sum_{n=1}^{\infty} \cos (n \pi)$ : find partial sums $S_{k}$ , check convergence, and use a test.

See Solution

Problem 22023

Differentiate and optimize: $5 x^{2}+4 x-16$ . Find $x=$ , with a (max or min) value of. Show workings to 1.d.p.

See Solution

Problem 22024

Show that the function $f(x)=x^{3}+4 x^{2}-8 x-10$ has a zero between $a=-6$ and $b=-5$ using the Intermediate Value Theorem.

See Solution

Problem 22025

Calculate the average value of $f(x)=x^{2}-15$ over the interval from $x=0$ to $x=6$ : $\square$ .

See Solution

Problem 22026

A ball is thrown horizontally at $20 \mathrm{~m/s}$ from 42 m high. How long until it hits the ground?

See Solution

Problem 22027

Find the arc length of $y=\ln \left(x-\sqrt{x^{2}-1}\right)$ for $1 \leq x \leq \sqrt{2}$ .

See Solution

Problem 22028

Calculate the arc length of $y=\ln \left(x-\sqrt{x^{2}-1}\right)$ for $1 \leq x \leq \sqrt{2}$ . Simplify your answer.

See Solution

Problem 22029

Find the volume $V$ of the solid formed by rotating the area between $y=x^{3}$ , $y=1$ , and $x=2$ around $y=-4$ .

See Solution

Problem 22030

Calculate the arc length of $y=\ln \left(x-\sqrt{x^{2}-1}\right)$ for $1 \leq x \leq \sqrt{2}$ .

See Solution

Problem 22031

Find the max and min of $y=\frac{1}{2} \sin (2 x)+3 \cos (x)+x$ for $0<x<2 \pi$ . [4 marks]

See Solution

Problem 22032

Find the partial fraction expansion of $a_{n}=\frac{3}{n^{2}+3n}$ and analyze the convergence of the series $\sum_{n=1} a_{n}$ .

See Solution

Problem 22033

Find the integral for the arc length of $y=\frac{3}{x}$ from $x=1$ to $x=10$ and simplify it. Options: A. $\int_{1}^{10} \sqrt{1+\frac{9}{x^{4}}} dx$ , B. $\int_{1}^{10} \sqrt{1-\frac{3}{x^{4}}} dx$ , C. $\int_{1}^{10}-\frac{3}{x^{2}} dx$ , D. $\int_{1}^{10} \sqrt{1+\frac{9}{x^{2}}} dx$ .

See Solution

Problem 22034

(11) Expand $a_{n}=\frac{3}{n^{2}+3n}$ using partial fractions and analyze its convergence. (12) Check if $\sum_{n=0}^{\infty} a_{n}$ converges with $a_{n+1}=\frac{a_{n}}{3}$ , starting from $a_{0}=10$ . (13) Use the ratio or root test to check convergence for: (a) $\sum_{n=1}^{\infty} \frac{10^{n}}{n!}$ (b) $\sum_{n=1}^{\infty}\left(\frac{n^{2}+1}{2n^{2}+1}\right)^{n}$ (c) $\sum_{n=1}^{\infty} \frac{3^{n}}{5^{n}+n}$ (d) $\sum_{n=2}^{\infty}\left(\frac{n+1}{n}\right)^{n^{2}}$ (e) $\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{n}}$ (14) Determine convergence or divergence for: (a) $\sum_{n=1}^{\infty} \frac{\tan(1/n)}{n}$ (b) $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2n-1)}{5^{n} n!}$ (c) $\sum_{n=1}^{\infty}(\sqrt[n]{2}-1)^{n}$

See Solution

Problem 22035

Find the arc length of the curve $y=\frac{4}{x}$ for $5 \leq x \leq 15$ using the integral: $\int_{5}^{15} \sqrt{1+\frac{16}{x^{4}}} dx$ . Approximate the length.

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Problem 22036

Find the average rate of change of the function $g(x)=-x^{2}-9 x+27$ over the interval $-9 \leq x \leq -1$ .

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Problem 22037

Ein Flugzeug bewegt sich gemäß der Funktion $f(t)=\frac{1}{6075}\left(t^{4}-\frac{140}{3} t^{3}+672 t^{2}-2880 t+34146\right)$ für $t \in [0; 25]$ Minuten.
a) Bestimmen Sie die minimale und maximale Flughöhe. b) Finden Sie die maximale Flughöhe in den ersten 4 Minuten. c) Was ist der Grenzwert der Höhe für $t \to \infty$ und was bedeutet das für den Definitionsbereich?

See Solution

Problem 22038

Evaluate the integrals $\int_{a}^{b} \sqrt{1+64 x^{8}} d x$ and $\int_{a}^{b} \sqrt{1+225 \cos ^{2}(3 x)} d x$ . Choose correct answers for their antiderivatives.

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Problem 22039

Berechne die Steigung von $f$ an $x_0$ mit der h-Methode für: a) $f(x)=x^{2}-x, x_{0}=1$ b) $f(x)=2 x^{2}+1, x_{0}=-2$ c) $f(x)=3 x+2, x_{0}=2$

See Solution

Problem 22040

Determine convergence of $\sum_{n=0}^{\infty} a_{n}$ where $a_{n+1}=\frac{a_{n}}{3}$ and $a_{0}=10$ . Find its value if it converges.

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Problem 22041

Find the arc length of the line $y=5x+2$ on $[0,4]$ using calculus. Which integral represents this arc length?

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Problem 22042

Compare the lengths of the curves $y=1-x^{2}$ and $y=\cos (\pi x / 2)$ over $[-1,1]$ . Which is longer?

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Problem 22043

Find the average temperature from the function $T(t)=40+6 t-\frac{1}{3} t^{2}$ over 18 hours.

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Problem 22044

Estimate the balance of a savings account with daily deposits of \$12,000 at 8\% interest compounded continuously after 6 years.

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Problem 22045

Determine convergence of these series using the ratio or root test: (a) $\sum_{n=1}^{\infty} \frac{10^{n}}{n !}$ , (b) $\sum_{n=1}^{\infty}\left(\frac{n^{2}+1}{2 n^{2}+1}\right)^{n}$ , (c) $\sum_{n=1}^{\infty} \frac{3^{n}}{5^{n}+n}$ , (d) $\sum_{n=2}^{\infty}\left(\frac{n+1}{n}\right)^{n^{2}}$ , (e) $\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{n}}$ .

See Solution

Problem 22046

Estimate the balance of a savings account with daily deposits of \ $19,000 at$ 8\%$ interest compounded continuously after 5 years.

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Problem 22047

Calculate the average rate of change of $f(x)=2x^{2}+6$ for these intervals: (a) 3 to 5, (b) 1 to 3, (c) 3 to 6.

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Problem 22048

Determine if the following series converge or diverge: (a) $\sum_{n=1}^{\infty} \frac{\tan (1 / n)}{n}$ , (b) $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{5^{n} n !}$ , (c) $\sum_{n=1}^{\infty}(\sqrt[n]{2}-1)^{n}$ .

See Solution

Problem 22049

Determine the convergence or divergence of these series: (a) $\sum_{n=1}^{\infty} \frac{\tan (1 / n)}{n}$ (b) $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{5^{n} n !}$ (c) $\sum_{n=1}^{\infty}(\sqrt[n]{2}-1)^{n}$ (d) $\sum_{n=1}^{\infty} \frac{(2 n)^{n}}{n^{2 n}}$ (e) $\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{\ln n}}$

See Solution

Problem 22050

Evaluate the integral from 1 to x: $\int_{1}^{x} y^{b-1} d y$ .

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Problem 22051

Find the derivative of $f(x)=\cos^{2}(5x^{2}-9)$ with respect to $x$ .

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Problem 22052

A thermometer at $79^{\circ} \mathrm{F}$ cools to $57^{\circ} \mathrm{F}$ in a cold room at $35^{\circ} \mathrm{F}$ . How long?

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Problem 22053

Find the tangent line equations for the curve $x=t^{2}, y=t^{3}-t$ at points $(4,6)$ and $(1,0)$ . Why two at $(1,0)$ ?

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Problem 22054

Differentiate the function $f(x)=2 e^{5\left(x^{2}+1\right)^{2}}$ .

See Solution

Problem 22055

Find the derivative of the function $y=10^{6 x+1}$ .

See Solution

Problem 22056

Differentiate the function $f(x)=\ln(9-8x^{2})$ with respect to $x$ .

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Problem 22057

Differentiate $y^{\prime}$ for $y=(8 x)^{\ln 8 x}$ using logarithmic differentiation.

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Problem 22058

Evaluate the limit $\lim _{x \rightarrow 0} \frac{e^{x}-1}{4 x}=\square$ .

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Problem 22059

Find the antiderivative $F(t)$ of $f(t)=6 \sec ^{2}(t)-2 t^{3}$ with $F(0)=0$ . Calculate $F(4)$ .

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Problem 22060

Find the max and min of $s=\sin t-\cos t$ for $0<t<2\pi$ and when they occur. [4 marks]

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Problem 22061

Find the limit: $\lim _{x \rightarrow 0} \frac{-x-\ln (1-x)}{10 x^{2}}$ and simplify your answer.

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Problem 22062

Find the limit using Taylor series: $\lim _{x \rightarrow 0} \frac{(1+x)-e^{x}}{5 x^{2}}=\square$

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Problem 22063

Find the limit: $\lim _{t \rightarrow 0} \frac{3 t^{2}+6 \cos t-6}{t^{4}}=\square($ Simplify your answer. $)$

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Problem 22064

Find the population size at $t=5.2$ given a growth rate of $3\%$ and $N(5)=100$ . The answer is $\square$ .

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Problem 22065

If the artery radius $\mathrm{R}$ is accurate to $2\%$ , how accurate is the speed $v(R)=C R^{2}$ ? Use linear approximations.

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Problem 22066

Find the limit: $\lim _{x \rightarrow \infty} 5 x \sin \frac{1}{7 x} = \square$ (Enter an integer or simplified fraction.)

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Problem 22067

Find the speed error of blood flow $v(R)=\mathrm{CR}^{2}$ with a $2\%$ radius accuracy. Which error formula is correct?

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Problem 22068

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\ln (1+x)-x+\frac{x^{2}}{2}}{4 x^{3}}=\square($ Simplify your answer.)

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Problem 22069

Determine the fetal growth rate $\frac{\mathrm{dH}}{\mathrm{dt}}$ for $H=-30.03+1.262 t^{2}-0.2532 t^{2} \log t$ at $t=8$ and $t=36$ weeks.

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Problem 22070

The area of a circle grows at 344 sq ft/min. Find the circumference change rate when the radius is 9. Round to three decimals.

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Problem 22071

A cube's volume decreases at 1678 ft³/s. When the side is 9 ft, find the surface area change rate. Round to three decimals.

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Problem 22072

Determine if the series $\sum_{n=1}^{\infty} \frac{\arctan n}{n}$ converges or diverges.

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Problem 22073

Find the rate of change of the surface area of a cube with volume 186 m³ decreasing at 511 m³/min. Round to three decimals.

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Problem 22074

A cube's volume decreases by 511 m³/min. When its volume is 186 m³, find the surface area change rate. Round to three decimals.

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Problem 22075

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{7 \tan ^{-1} x-7 x+\frac{7}{3} x^{3}}{x^{5}}$ using Taylor series.

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Problem 22076

Find the particle's maximum speed on the interval $0 \leq t \leq 2$ for the position function $y(t)=6t-2t^3+10$ .

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Problem 22077

Given the function $f(x) = -0.5x^4 + 3x^2$ , find intervals of positive/negative concavity and any inflection points.

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Problem 22078

Invest \$3000 at a 5% annual interest rate. How much will it be after 9 years with continuous compounding?

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Problem 22079

Show that $P(x)=3 x^{4}-3 x^{2}+2 x-1$ has a zero between 0.5 and 1 using the intermediate value theorem. Calculate $P(0.5)$ .

See Solution

Problem 22080

Find the absolute maximum of the function $y=x^{2}-2 x+1$ .

See Solution

Problem 22081

Find $g^{\prime}(1)$ for $g(x)=4 x^{3}-6 x^{2}+8 x+\sqrt{10}$ .

See Solution

Problem 22082

Find $g^{\prime}(1)$ for $g(x)=4 x^{3}-6 x^{2}+8 x+\sqrt{10}$ . Options: $6$ , $6+\sqrt{10}$ , $8$ , $8+\sqrt{10}$ .

See Solution

Problem 22083

Verify if the function $P(x)=x^{5}-3 x^{3}+2$ has a real zero between -1.8 and -1.9 using the intermediate value theorem.

See Solution

Problem 22084

Find the instantaneous rate of change of $g(x)=\lim _{h \rightarrow 0} \frac{e^{x+h}-e^{x}}{h}$ at $x=2$ .

See Solution

Problem 22085

Find the future value of an investment of \$18,000 for 14 years with continuous compounding at: (a) 3\%, (b) 4\%, (c) 5.5\%.

See Solution

Problem 22086

Find the exponential growth function for a world population of 7.5 billion with a growth rate of $1.06\%$ . Then, answer:
a) What is $P(t)$ ? b) World population in 2028? c) Year it reaches 14 billion? d) Doubling time?

See Solution

Problem 22087

Find the intervals where the function $g(x)=-x^{4}+2 x^{2}-1$ is both concave up and decreasing.

See Solution

Problem 22088

Find the intervals where the function $g(x)=-x^{4}+2 x^{2}-1$ is both concave up and decreasing.

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Problem 22089

Identify which function satisfies the Mean Value Theorem on $[0,5]$ : (A) $f(x)=\frac{x-3}{x+3}$ (B) $f(x)=(x-1)^{\frac{2}{3}}$ (C) $f(x)=\frac{x+3}{x-3}$ (D) $f(x)=|x-4|$

See Solution

Problem 22090

Graph $f(x)=-0.5 \cdot x^{4}+3 x^{2}$ . Find intervals of positive/negative concavity and any inflection points.

See Solution

Problem 22091

Calculate the following: (a) $5^{3}$ , (b) $\Delta 5^{\underline{n}}$ , (c) $\Delta\left(\sum_{k=1}^{n} \sin (k !)\right)$ , (d) What continuous calculus concept relates to (c)?

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Problem 22092

Find relative extrema of $g$ given $g^{\prime}(x)=(5-x) x^{-3}$ for $x>0$ . Justify your conclusions.

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Problem 22093

The derivative of $x^{e}$ is $x^{e} \ln (x)$ . Is this statement True or False?

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Problem 22094

Find the limit: $\lim _{x \rightarrow \infty} 2 x^{3}\left(e^{-\frac{2}{x^{3}}}-1\right)$ .

See Solution

Problem 22095

Given the parametric equations $x=-4+\sec t$ and $y=5+2 \tan t$ , find the first and second derivatives.

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Problem 22096

If $f$ is differentiable and $f^{\prime}(c)=0$ , is $x=c$ a critical number for $f$ ? True or False?

See Solution

Problem 22097

Find the absolute maximum and minimum of the function $f(x)=e^{x} \sin x$ on the interval $[0,2 \pi]$ .

See Solution

Problem 22098

Is it true that a continuous function $g$ on $(-\pi, \pi)$ has an absolute minimum in that interval? True or False?

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Problem 22099

Find the rectangle's dimensions with max area, one corner at (0,0) and opposite at $(x, y)$ on $y=\frac{3}{1+4 x^{2}}$ , $0 \leq x \leq 3$ .

See Solution

Problem 22100

Find the differential $d y$ for $y=\sqrt{9+7 x^{2}}$ at $x=1$ with $d x=0.1$ . Provide the answer as a decimal.

See Solution

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Calculus

Problem 22001

Find where the function f(x)=3x2+(x−4)2/3+1xf(x)=3 x^{2}+(x-4)^{2 / 3}+\frac{1}{x}f(x)=3x2+(x−4)2/3+x1​ is continuous.

Problem 22002

Find the tangent line equation for f(x)=25x2+1f(x)=\frac{25}{x^{2}+1}f(x)=x2+125​ at x=2x=2x=2.

Problem 22003

Find where the function f(x)=3x2+(x−4)23+1xf(x) = 3x^{2} + (x-4)^{\frac{2}{3}} + \frac{1}{x}f(x)=3x2+(x−4)32​+x1​ is continuous.

Problem 22004

Inflate a balloon at 16 in³/s. Find the radius increase rate when it’s 3 inches. Give exact or rounded answer with units.

Problem 22005

Evaluate the integral from 0 to 1 of (2+x2)(2+x^{2})(2+x2) with respect to xxx.

Problem 22006

A rock thrown up on the moon at 32 m/s has height s(t)=32t−0.8t2s(t)=32t-0.8t^2s(t)=32t−0.8t2. Solve for velocity, time to max height, and max height.

Problem 22007

Find the drug concentration C(9)\mathrm{C}(9)C(9) after 9 minutes using C(t)=0.07(1−e−0.2t)C(t)=0.07(1-e^{-0.2 t})C(t)=0.07(1−e−0.2t). Round to three decimals.

Problem 22008

Find the absolute max and min of f(x)=x3+6x2−2f(x)=x^{3}+6x^{2}-2f(x)=x3+6x2−2 on the interval −5≤x≤1-5 \leq x \leq 1−5≤x≤1.

Problem 22009

Find the absolute max and min of f(x)=x3+6x2−2f(x) = x^{3} + 6x^{2} - 2f(x)=x3+6x2−2 on [−5,1][-5, 1][−5,1]. Min = -30; Max = 2.

Problem 22010

Find the elasticity of demand for D(p)=155−p2D(p) = 155 - p^{2}D(p)=155−p2 at p=11p = 11p=11 and classify it as elastic, inelastic, or unit elastic.

Problem 22011

Find the production level xxx that minimizes the average cost given c(x)=5x3−35x2+5,547xc(x)=5x^{3}-35x^{2}+5,547xc(x)=5x3−35x2+5,547x.

Problem 22012

Find the derivative of e4x2+3xy+y2e^{4 x^{2}+3 x y+y^{2}}e4x2+3xy+y2 with respect to xxx.

Problem 22013

Problem 22014

Find the partial derivative ∂∂yx2ln⁡(2xy+3x+y3)\frac{\partial}{\partial y} x^{2} \ln \left(2 x y+3 x+y^{3}\right)∂y∂​x2ln(2xy+3x+y3).

Problem 22015

Find the derivative ∂∂xx2ln⁡(xy+3x+y2)\frac{\partial}{\partial x} x^{2} \ln (x y + 3 x + y^{2})∂x∂​x2ln(xy+3x+y2).

Problem 22016

Find the arc length of the curve y=x66+116x4y=\frac{x^{6}}{6}+\frac{1}{16 x^{4}}y=6x6​+16x41​ on [1,3][1,3][1,3]. Length: □\square□.

Problem 22017

112 rabbits are released, growing to 1008 in 2 years. What will the population be after another 9 months? Round to nearest rabbit.

Problem 22018

Find the volume VVV of the solid formed by rotating the area between x=8y2x=8y^2x=8y2 and x=8x=8x=8 around the line x=8x=8x=8.

Problem 22019

Find the arc length of the curve y=x44+18x2y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}}y=4x4​+8x21​ on the interval [1,4][1,4][1,4].

Problem 22020

Find the exponential growth rate of a bacteria culture that doubles in 8hr8 \mathrm{hr}8hr. Options: a) 5.5%5.5 \%5.5% b) 8.7%8.7 \%8.7% c) 3.8%3.8 \%3.8% d) 4.2%4.2 \%4.2%

Problem 22021

Find the tangent line equation to y=f(x)y=f(x)y=f(x) at x=ex=ex=e, where f(x)=x−6ln⁡xf(x)=x-6 \ln xf(x)=x−6lnx.

Problem 22022

Evaluate the series ∑n=1∞cos⁡(nπ)\sum_{n=1}^{\infty} \cos (n \pi)∑n=1∞​cos(nπ): find partial sums SkS_{k}Sk​, check convergence, and use a test.

Problem 22023

Differentiate and optimize: 5x2+4x−165 x^{2}+4 x-165x2+4x−16. Find x=x=x=, with a (max or min) value of. Show workings to 1.d.p.

Problem 22024

Show that the function f(x)=x3+4x2−8x−10f(x)=x^{3}+4 x^{2}-8 x-10f(x)=x3+4x2−8x−10 has a zero between a=−6a=-6a=−6 and b=−5b=-5b=−5 using the Intermediate Value Theorem.

Problem 22025

Calculate the average value of f(x)=x2−15f(x)=x^{2}-15f(x)=x2−15 over the interval from x=0x=0x=0 to x=6x=6x=6: □\square□.

Problem 22026

A ball is thrown horizontally at 20 m/s20 \mathrm{~m/s}20 m/s from 42 m high. How long until it hits the ground?

Problem 22027

Find the arc length of y=ln⁡(x−x2−1)y=\ln \left(x-\sqrt{x^{2}-1}\right)y=ln(x−x2−1​) for 1≤x≤21 \leq x \leq \sqrt{2}1≤x≤2​.

Problem 22028

Calculate the arc length of y=ln⁡(x−x2−1)y=\ln \left(x-\sqrt{x^{2}-1}\right)y=ln(x−x2−1​) for 1≤x≤21 \leq x \leq \sqrt{2}1≤x≤2​. Simplify your answer.

Problem 22029

Find the volume VVV of the solid formed by rotating the area between y=x3y=x^{3}y=x3, y=1y=1y=1, and x=2x=2x=2 around y=−4y=-4y=−4.

Problem 22030

Calculate the arc length of y=ln⁡(x−x2−1)y=\ln \left(x-\sqrt{x^{2}-1}\right)y=ln(x−x2−1​) for 1≤x≤21 \leq x \leq \sqrt{2}1≤x≤2​.

Problem 22031

Find the max and min of y=12sin⁡(2x)+3cos⁡(x)+xy=\frac{1}{2} \sin (2 x)+3 \cos (x)+xy=21​sin(2x)+3cos(x)+x for 0<x<2π0<x<2 \pi0<x<2π. [4 marks]

Problem 22032

Find the partial fraction expansion of an=3n2+3na_{n}=\frac{3}{n^{2}+3n}an​=n2+3n3​ and analyze the convergence of the series ∑n=1an\sum_{n=1} a_{n}∑n=1​an​.

Problem 22033

Problem 22034

Problem 22035

Find the arc length of the curve y=4xy=\frac{4}{x}y=x4​ for 5≤x≤155 \leq x \leq 155≤x≤15 using the integral: ∫5151+16x4dx\int_{5}^{15} \sqrt{1+\frac{16}{x^{4}}} dx∫515​1+x416​​dx. Approximate the length.

Problem 22036

Find the average rate of change of the function g(x)=−x2−9x+27g(x)=-x^{2}-9 x+27g(x)=−x2−9x+27 over the interval −9≤x≤−1-9 \leq x \leq -1−9≤x≤−1.

Problem 22037

Problem 22038

Evaluate the integrals ∫ab1+64x8dx\int_{a}^{b} \sqrt{1+64 x^{8}} d x∫ab​1+64x8​dx and ∫ab1+225cos⁡2(3x)dx\int_{a}^{b} \sqrt{1+225 \cos ^{2}(3 x)} d x∫ab​1+225cos2(3x)​dx. Choose correct answers for their antiderivatives.

Problem 22039

Berechne die Steigung von fff an x0x_0x0​ mit der h-Methode für: a) f(x)=x2−x,x0=1f(x)=x^{2}-x, x_{0}=1f(x)=x2−x,x0​=1 b) f(x)=2x2+1,x0=−2f(x)=2 x^{2}+1, x_{0}=-2f(x)=2x2+1,x0​=−2 c) f(x)=3x+2,x0=2f(x)=3 x+2, x_{0}=2f(x)=3x+2,x0​=2

Problem 22040

Determine convergence of ∑n=0∞an\sum_{n=0}^{\infty} a_{n}∑n=0∞​an​ where an+1=an3a_{n+1}=\frac{a_{n}}{3}an+1​=3an​​ and a0=10a_{0}=10a0​=10. Find its value if it converges.

Problem 22041

Find the arc length of the line y=5x+2y=5x+2y=5x+2 on [0,4][0,4][0,4] using calculus. Which integral represents this arc length?

Problem 22042

Find where the function $f(x)=3 x^{2}+(x-4)^{2 / 3}+\frac{1}{x}$ is continuous.

Find the tangent line equation for $f(x)=\frac{25}{x^{2}+1}$ at $x=2$ .

Find where the function $f(x) = 3x^{2} + (x-4)^{\frac{2}{3}} + \frac{1}{x}$ is continuous.

Evaluate the integral from 0 to 1 of $(2+x^{2})$ with respect to $x$ .

A rock thrown up on the moon at 32 m/s has height $s(t)=32t-0.8t^2$ . Solve for velocity, time to max height, and max height.

Find the drug concentration $\mathrm{C}(9)$ after 9 minutes using $C(t)=0.07(1-e^{-0.2 t})$ . Round to three decimals.

Find the absolute max and min of $f(x)=x^{3}+6x^{2}-2$ on the interval $-5 \leq x \leq 1$ .

Find the absolute max and min of $f(x) = x^{3} + 6x^{2} - 2$ on $[-5, 1]$ . Min = -30; Max = 2.

Find the elasticity of demand for $D(p) = 155 - p^{2}$ at $p = 11$ and classify it as elastic, inelastic, or unit elastic.

Find the production level $x$ that minimizes the average cost given $c(x)=5x^{3}-35x^{2}+5,547x$ .

Find the derivative of $e^{4 x^{2}+3 x y+y^{2}}$ with respect to $x$ .

Find the partial derivative $\frac{\partial}{\partial y} x^{2} \ln \left(2 x y+3 x+y^{3}\right)$ .

Find the derivative $\frac{\partial}{\partial x} x^{2} \ln (x y + 3 x + y^{2})$ .

Find the arc length of the curve $y=\frac{x^{6}}{6}+\frac{1}{16 x^{4}}$ on $[1,3]$ . Length: $\square$ .

Find the volume $V$ of the solid formed by rotating the area between $x=8y^2$ and $x=8$ around the line $x=8$ .

Find the arc length of the curve $y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}}$ on the interval $[1,4]$ .

Find the exponential growth rate of a bacteria culture that doubles in $8 \mathrm{hr}$ . Options: a) $5.5 \%$ b) $8.7 \%$ c) $3.8 \%$ d) $4.2 \%$

Find the tangent line equation to $y=f(x)$ at $x=e$ , where $f(x)=x-6 \ln x$ .

Evaluate the series $\sum_{n=1}^{\infty} \cos (n \pi)$ : find partial sums $S_{k}$ , check convergence, and use a test.

Differentiate and optimize: $5 x^{2}+4 x-16$ . Find $x=$ , with a (max or min) value of. Show workings to 1.d.p.

Show that the function $f(x)=x^{3}+4 x^{2}-8 x-10$ has a zero between $a=-6$ and $b=-5$ using the Intermediate Value Theorem.

Calculate the average value of $f(x)=x^{2}-15$ over the interval from $x=0$ to $x=6$ : $\square$ .

A ball is thrown horizontally at $20 \mathrm{~m/s}$ from 42 m high. How long until it hits the ground?

Find the arc length of $y=\ln \left(x-\sqrt{x^{2}-1}\right)$ for $1 \leq x \leq \sqrt{2}$ .

Calculate the arc length of $y=\ln \left(x-\sqrt{x^{2}-1}\right)$ for $1 \leq x \leq \sqrt{2}$ . Simplify your answer.

Find the volume $V$ of the solid formed by rotating the area between $y=x^{3}$ , $y=1$ , and $x=2$ around $y=-4$ .

Calculate the arc length of $y=\ln \left(x-\sqrt{x^{2}-1}\right)$ for $1 \leq x \leq \sqrt{2}$ .

Find the max and min of $y=\frac{1}{2} \sin (2 x)+3 \cos (x)+x$ for $0<x<2 \pi$ . [4 marks]

Find the partial fraction expansion of $a_{n}=\frac{3}{n^{2}+3n}$ and analyze the convergence of the series $\sum_{n=1} a_{n}$ .

Find the arc length of the curve $y=\frac{4}{x}$ for $5 \leq x \leq 15$ using the integral: $\int_{5}^{15} \sqrt{1+\frac{16}{x^{4}}} dx$ . Approximate the length.

Find the average rate of change of the function $g(x)=-x^{2}-9 x+27$ over the interval $-9 \leq x \leq -1$ .

Evaluate the integrals $\int_{a}^{b} \sqrt{1+64 x^{8}} d x$ and $\int_{a}^{b} \sqrt{1+225 \cos ^{2}(3 x)} d x$ . Choose correct answers for their antiderivatives.

Berechne die Steigung von $f$ an $x_0$ mit der h-Methode für: a) $f(x)=x^{2}-x, x_{0}=1$ b) $f(x)=2 x^{2}+1, x_{0}=-2$ c) $f(x)=3 x+2, x_{0}=2$

Determine convergence of $\sum_{n=0}^{\infty} a_{n}$ where $a_{n+1}=\frac{a_{n}}{3}$ and $a_{0}=10$ . Find its value if it converges.

Find the arc length of the line $y=5x+2$ on $[0,4]$ using calculus. Which integral represents this arc length?

Compare the lengths of the curves $y=1-x^{2}$ and $y=\cos (\pi x / 2)$ over $[-1,1]$ . Which is longer?

Find the average temperature from the function $T(t)=40+6 t-\frac{1}{3} t^{2}$ over 18 hours.

Estimate the balance of a savings account with daily deposits of \ $19,000 at$ 8\%$ interest compounded continuously after 5 years.

Calculate the average rate of change of $f(x)=2x^{2}+6$ for these intervals: (a) 3 to 5, (b) 1 to 3, (c) 3 to 6.

Evaluate the integral from 1 to x: $\int_{1}^{x} y^{b-1} d y$ .

Find the derivative of $f(x)=\cos^{2}(5x^{2}-9)$ with respect to $x$ .

A thermometer at $79^{\circ} \mathrm{F}$ cools to $57^{\circ} \mathrm{F}$ in a cold room at $35^{\circ} \mathrm{F}$ . How long?

Find the tangent line equations for the curve $x=t^{2}, y=t^{3}-t$ at points $(4,6)$ and $(1,0)$ . Why two at $(1,0)$ ?

Differentiate the function $f(x)=2 e^{5\left(x^{2}+1\right)^{2}}$ .

Find the derivative of the function $y=10^{6 x+1}$ .

Differentiate the function $f(x)=\ln(9-8x^{2})$ with respect to $x$ .

Differentiate $y^{\prime}$ for $y=(8 x)^{\ln 8 x}$ using logarithmic differentiation.

Evaluate the limit $\lim _{x \rightarrow 0} \frac{e^{x}-1}{4 x}=\square$ .

Find the antiderivative $F(t)$ of $f(t)=6 \sec ^{2}(t)-2 t^{3}$ with $F(0)=0$ . Calculate $F(4)$ .

Find the max and min of $s=\sin t-\cos t$ for $0<t<2\pi$ and when they occur. [4 marks]

Find the limit: $\lim _{x \rightarrow 0} \frac{-x-\ln (1-x)}{10 x^{2}}$ and simplify your answer.

Find the limit using Taylor series: $\lim _{x \rightarrow 0} \frac{(1+x)-e^{x}}{5 x^{2}}=\square$

Find the limit: $\lim _{t \rightarrow 0} \frac{3 t^{2}+6 \cos t-6}{t^{4}}=\square($ Simplify your answer. $)$

Find the population size at $t=5.2$ given a growth rate of $3\%$ and $N(5)=100$ . The answer is $\square$ .

If the artery radius $\mathrm{R}$ is accurate to $2\%$ , how accurate is the speed $v(R)=C R^{2}$ ? Use linear approximations.

Find the limit: $\lim _{x \rightarrow \infty} 5 x \sin \frac{1}{7 x} = \square$ (Enter an integer or simplified fraction.)

Find the speed error of blood flow $v(R)=\mathrm{CR}^{2}$ with a $2\%$ radius accuracy. Which error formula is correct?

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\ln (1+x)-x+\frac{x^{2}}{2}}{4 x^{3}}=\square($ Simplify your answer.)

Determine the fetal growth rate $\frac{\mathrm{dH}}{\mathrm{dt}}$ for $H=-30.03+1.262 t^{2}-0.2532 t^{2} \log t$ at $t=8$ and $t=36$ weeks.

Determine if the series $\sum_{n=1}^{\infty} \frac{\arctan n}{n}$ converges or diverges.