Calculus

Problem 31101

Find the tangent line approximation for $f(3.15)$ given $f(3)=-2$ and $f'(3)=4$ .

See Solution

Problem 31102

Approximate $f(-3.85)$ using the tangent line at $x=-4$ where $f(-4)=-8$ and $f^{\prime}(x)=-x-6$ .

See Solution

Problem 31103

Find the tangent line at $x=-3$ for $f$ with $f(-3)=0$ and $f^{\prime}(x)=-5x-6$ . Estimate $f(-2.85)$ .

See Solution

Problem 31104

Calcule a taxa média de variação de $f(x)$ no intervalo $3 \leq x \leq 4$ usando os pontos $(3,-1)$ e $(4,1)$ .

See Solution

Problem 31105

Find the average rate of change of $g(x)=-x^{2}+2 x+8$ from $x=-3$ to $x=3$ .

See Solution

Problem 31106

The town's population is 20000 and decreases by $1.6\%$ per year. How long until it reaches 15000?

See Solution

Problem 31107

Solve the ODE $\frac{d z}{d x}-8 x z=8 x^{2}-1$ with initial condition $z(1)=e-1$ .

See Solution

Problem 31108

Integrate the following: 1. $\int x \sin \frac{x}{2} d x$ , 3. $\int t^{2} \cos t d t$ , 5. $\int_{1}^{2} x \ln x d x$ , 7. $\int x e^{x} d x$ , 9. $\int x^{2} e^{-x} d x$ , 11. $\int \tan ^{-1} y d y$ , 13. $\int x \sec ^{2} x d x$ , 15. $\int x^{3} e^{x} d x$ .

See Solution

Problem 31109

Find the slope of the tangent line to $f^{-1}(x)$ at (0,1) where $f(x)=2-x-x^{3}$ . Provide the exact answer.

See Solution

Problem 31110

Find the average rate of change of horsepower, $H(s)=0.003 s^{2}+0.07 s-0.027$ , as speed increases from 80 mph to 100 mph.

See Solution

Problem 31111

Find the average rate of change of $f(x) = 2 \sqrt{x} + 5$ on the interval $[4, 9]$ .

See Solution

Problem 31112

Prove that for all $x, y \in \mathrm{R}$ , the inequality holds: $\left|\arctan(x^3) - \arctan(y^3)\right| \leq M|x-y|$ where $M=\sup_{\operatorname{tat}} \frac{3t^2}{1+t^6}$ .

See Solution

Problem 31113

A store's profit function for fans is $P(x)=-x^{2}+39 x-240$ . Find the average profit change for price increases from \$10 to \$11 and from \$20 to \$21.

See Solution

Problem 31114

Find the rate of change of $x$ when $x=\frac{\pi}{2}$ , given $y=2 e^{\cos (x)}$ and $\frac{dy}{dt}=5$ .

See Solution

Problem 31115

Find the expression for $\frac{f(x+h)-f(x)}{h}$ where $f(x)=4 x^{2}-9$ .

See Solution

Problem 31116

Evaluate the limit:
$\lim _{(x, y) \rightarrow(8,7)} \frac{x-y-1}{\sqrt{x-y}-1}=$

See Solution

Problem 31117

1. Find $\int \frac{1-x}{\sqrt{1-x^{2}}} d x$
2. Evaluate $\int \frac{e^{-\cot z}}{\sin ^{2} z} d z$
3. Compute $\int \frac{d z}{e^{z}+e^{-z}}$
4. Solve $\int_{-1}^{0} \frac{4 d x}{1+(2 x+1)^{2}}$
5. Determine $\int \frac{d t}{1-\sec t}$
6. Find $\int_{0}^{\pi / 4} \frac{1+\sin \theta}{\cos ^{2} \theta} d \theta$

See Solution

Problem 31118

Prove that for all $x, y \in \mathrm{R}$ , the inequality $|\arctan(x^3) - \arctan(y^3)| \leq M|x-y|$ holds, where $M = \sup_{t \in x} \frac{3t^2}{1+t^6}$ .

See Solution

Problem 31119

Find the limit: $\lim _{x \rightarrow-3} 7 x^{2}$ . Complete the table for values around $-3$ .

See Solution

Problem 31120

Find the limit: $\lim _{x \rightarrow 1}\left(-7 x^{3}\right)$ . Fill in the table for $f(x)=-7 x^{3}$ at values near 1.

See Solution

Problem 31121

Find the displacement of a sled starting from rest, accelerating at $50 \mathrm{~m/s^2}$ for $5 \mathrm{~s}$ .

See Solution

Problem 31122

Find the limit: $\lim _{x \rightarrow 3} 9 x^{3}$ . Fill in the table for $f(x)=9 x^{3}$ at values around $x=3$ .

See Solution

Problem 31123

Find the limit: $\lim _{x \rightarrow 3}\left(-3 x^{2}\right)$ by completing the table for $f(x)=-3 x^{2}$ .

See Solution

Problem 31124

Find the limit: $\lim _{x \rightarrow-2}(-4 x^{2})$ using a table of values around $x = -2$ .

See Solution

Problem 31125

Find the limit: $\lim _{x \rightarrow 3}\left(-3 x^{2}\right)$ . Complete the table for $f(x)=-3 x^{2}$ .

See Solution

Problem 31126

Find the derivative $f'(x)$ for the function $f(x)=-8 \ln x + 5 x^2 - 12$ .

See Solution

Problem 31127

Find the derivative $f^{\prime}(x)$ for the function $f(x)=3 e^{x}+2 x-5 \ln x$ .

See Solution

Problem 31128

Differentiate the function $f(x)=-3+2x+2e^{x}$ . Find $f^{\prime}(x)$ .

See Solution

Problem 31129

Find the derivative of $f(x)=\ln x^{10}$ with respect to $x$ . What is $\frac{df}{dx}$ ?

See Solution

Problem 31130

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\ln x^{9}+7 \ln x$ .

See Solution

Problem 31131

Find the limit as $x$ approaches -7: $\lim _{x \rightarrow-7} x = \square$ or it does not exist.

See Solution

Problem 31132

Find the second order Taylor polynomial of $f$ at $x_0=0$ given $f(0)=0$ and $x+f(x)+\ln(1+f(x))=0$ .

See Solution

Problem 31133

Find the limit: $\lim _{x \rightarrow 8}(6 x-4)$ . Choose A for a value or B if it doesn't exist.

See Solution

Problem 31134

Find the limit: $\lim _{x \rightarrow-3} 8 x^{4}$ . Is it an integer, fraction, or does it not exist?

See Solution

Problem 31135

Find the limit: $\lim _{x \rightarrow 1}-5$ . Choose A or B and fill in the answer.

See Solution

Problem 31136

Find the limit: $\lim _{x \rightarrow 4} \frac{x}{x+5}$ . A. Answer: $\square$ (integer or simplified fraction) B. Limit does not exist.

See Solution

Problem 31137

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{2}-36}{x-6}$ . Choose A (value) or B (does not exist).

See Solution

Problem 31138

Rewrite $f(x) = 15x + \ln(15) + \ln(x)$ using logarithm properties, then find $f^{\prime}(x)$ .

See Solution

Problem 31139

Given the function $f(x)=2 \cos ^{2}(x)-4 \sin (x)$ for $0 \leq x \leq 2 \pi$ , find:
(a) Intervals of increase and decrease. (b) Local min and max values. (c) Inflection points and intervals of concavity.

See Solution

Problem 31140

Find the limit as $h$ approaches 0 for $\frac{\sqrt{9 h+1}-1}{h}$ . Is it A. $\square$ or B. Does not exist?

See Solution

Problem 31141

Find the limit: $\lim _{x \rightarrow 3} \frac{5 x-15}{x-3}$ . Choose A (value) or B (limit does not exist).

See Solution

Problem 31142

Évaluer les limites suivantes : a) $\lim _{s \rightarrow 2^{+}}\left[\frac{1}{s-2}+\frac{4}{4-s^{2}}\right]$ b) $\lim _{x \rightarrow 0^{-}}\left[\frac{e^{2 x}}{\sin 5 x}-\frac{1}{\tan 5 x}\right]$

See Solution

Problem 31143

Find $f^{\prime}(x)$ for $f(x)=-6x+4$ and determine the tangent line slopes at $x=-4$ and $x=4$ .

See Solution

Problem 31144

Find the derivative of $f(x)=x^{2}-4x+8$ and the slope at $x=\frac{3}{2}$ and $x=2$ .

See Solution

Problem 31145

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin x}$ .

See Solution

Problem 31146

Find $\frac{d f^{-1}}{d x}$ at $x=f(3)$ if $\frac{d f(x)}{d x}=x^{2}-3 x+2$ for $x>1$ . Choices: a. 2 b. 3 c. none d. $\frac{1}{2}$ e. $\frac{1}{3}$

See Solution

Problem 31147

Find the slope and equation of the tangent line for $f(x)=2x^{2}-5x$ at the point (2,-2). Slope: $\square$ .

See Solution

Problem 31148

Find the limit: $\lim _{x \rightarrow 2}\left(3 x^{2}-8 x+7\right)^{2}$ . Is it A. $\square$ or B. does not exist?

See Solution

Problem 31149

Rewrite $f(x)=7 \ln \left(\frac{14}{x}\right)$ using logarithm properties, then find $f^{\prime}(x)$ . $f^{\prime}(x)=\square$

See Solution

Problem 31150

Find the derivative of $y=8 \log _{7} x$ . What is $\frac{d y}{d x}=\square$ ?

See Solution

Problem 31151

Find the limits: d) $\lim _{x \rightarrow 0^{+}}\left[\frac{1}{\operatorname{Arctan} x}-\frac{1}{x}\right]$ and e) $\lim _{\theta \rightarrow 0^{+}}\left[\csc \theta-\frac{\cos \sqrt{\theta}}{\sin \theta}\right]$ .

See Solution

Problem 31152

Find the limit: $\lim _{x \rightarrow 0^{+}}\left[\frac{1}{\operatorname{Arctan} x}-\frac{1}{x}\right]$ .

See Solution

Problem 31153

How much radium remains in 6836 if half-life is 1620 years, starting with 0.5 g? Also, find bacteria doubling time if 50 grow to 204800 in 3 min.

See Solution

Problem 31154

Evaluate $\lim_{x \rightarrow \infty} f(x)$ for $f(x)=\frac{2x^{2}+3x}{(x+1)(x+2)}$ to find horizontal asymptotes.

See Solution

Problem 31155

Find the derivative $\frac{d y}{d x}$ for the function $y=5^{x}$ . What is $\frac{d y}{d x}=\square$ ?

See Solution

Problem 31156

Find the derivative $\frac{d y}{d x}$ for the function $y=6^{x}+e^{4}$ . Type an exact answer.

See Solution

Problem 31157

Find the annual depreciation rate for the function $S(t)=700,000(0.8)^{t}$ after 1 and 5 years.

See Solution

Problem 31158

Model train position is given by $s(t)=2.5 t+17$ . Find speed, position after 4 sec, and time to reach 32 feet.

See Solution

Problem 31159

Untersuchen Sie die Funktion $f(x)=2 x \cdot e^{-x}$ auf Nullstellen, Extrema und Wendepunkte. Verhalten für $x \rightarrow \infty$ und $x \rightarrow -\infty$ ? Zeichnen Sie den Graphen für $-0.5 \leq x \leq 3$ und bestimmen Sie die Tangentengleichung im Ursprung.

See Solution

Problem 31160

Find the limit: $\lim _{\theta \rightarrow 0^{+}}\left[\csc \theta-\frac{\cos \sqrt{\theta}}{\sin \theta}\right]$ .

See Solution

Problem 31161

Find (A) the derivative of $F(x) S(x)$ without the product rule, and (B) $F^{\prime}(x) S^{\prime}(x)$ . Given $F(x)=x^{4}+1$ , $S(x)=x^{5}$ .

See Solution

Problem 31162

Finde die Kostenfunktion $K$ aus den Ableitungen $K^{\prime}(x)$ und beschreibe den Kostenverlauf.

See Solution

Problem 31163

Find $f(x+h) - f(x)$ and $\frac{f(x+h) - f(x)}{h}$ for the function $f(x) = 6 - x^{2}$ .

See Solution

Problem 31164

Evaluate the limits: $-x^{2} \leq \lim _{x \rightarrow 0} x^{2} \sin \left(\frac{1}{x}\right) \leq x^{2}$ .

See Solution

Problem 31165

A bucket with a mass of $2.30 \mathrm{~kg}$ is whirled in a circle of radius $1.40 \mathrm{~m}$ . Find the acceleration at the top ( $\mathrm{m} / \mathrm{s}^2$ ).

See Solution

Problem 31166

Find $\lim _{\theta \rightarrow 0} \frac{\sin (\theta)}{\theta}$ using the inequality $\cos (\theta) \leq \frac{\sin (\theta)}{\theta} \leq 1$ .

See Solution

Problem 31167

Find the limit: $\lim _{x \rightarrow+\infty}\left(1-\frac{5}{x}\right)^{3 x}$ .

See Solution

Problem 31168

Find the integral of $(\ln (x))^{2}$ with respect to $x$ .

See Solution

Problem 31169

Find the integral of the function $y e^{-y}$ with respect to $y$ : $\int y e^{-y} d y$ .

See Solution

Problem 31170

Calculate the integral: $\int(\pi-x) \cos (\pi x) \, dx$ .

See Solution

Problem 31171

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}}\left[\ln \left(\frac{1}{1-x}\right)\right]^{1-x}$ .

See Solution

Problem 31172

Aufgabe 1: Beschreibe die Graphen der Funktionen $f(x)=\frac{1}{x}$ , $f(x)=0,7^{x}$ , $f(x)=\frac{3}{x^{2}}$ , $f(x)=\frac{x^{2}+1}{x}$ .
Aufgabe 2: Bestimme den Definitionsbereich für $f(x)=\frac{x^{2}-4}{x-2}$ und $f(x)=\frac{1}{(x-3)^{2}}+1$ . Untersuche das Verhalten bei Definitionslücken.
Aufgabe 3: Tee hat $94^{\circ} \mathrm{C}$ , Umgebung $24^{\circ} \mathrm{C}$ . Temperaturunterschied verringert sich um $\frac{1}{3}$ pro Minute. a) Formel für Abkühlung. b) Wann ist Temperatur $10^{\circ}$ , $5^{\circ}$ , $1^{\circ}$ , $0,1^{\circ}$ von Raumtemperatur entfernt?

See Solution

Problem 31173

A student on a 50.0 m building kicks a stone at 4.00 m/s. How long until it hits the ground? Choices: (1) 5.10 s (2) 3.19 s (3) 10.2 s (4) 12.5 s

See Solution

Problem 31174

Find the slope and tangent line equation at point $P(4,36)$ for the curve $y=x^{2}+5x$ .

See Solution

Problem 31175

Find the volume of the solid formed by rotating the area in the first quadrant bounded by $y=\sec x$ , $x=\frac{\pi}{3}$ , and the axes around the $x$ -axis.

See Solution

Problem 31176

Zbadaj zbieżność szeregów:
(5 pkt) $\sum_{n=1}^{\infty} \sqrt[n]{\frac{n^{5}}{5^{n}}}$ ,
(5 pkt) $\sum_{n=1}^{\infty}\left(\frac{n+3}{n+2}\right)^{n^{2}}$ ,
(5 pkt) $\sum_{n=1}^{\infty} \frac{n^{n}}{n ! 5^{n}}$ ,
(5 pkt) $\sum_{n=1}^{\infty} \frac{n^{2}+5}{n^{4}+4 n}$ ,
(5 pkt) $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n+6}$ .

See Solution

Problem 31177

Oblicz granicę ciągu dla: $a_{n}=\frac{n^{2}}{5+10+15+\ldots+5 n}$ , $a_{n}=\sqrt{9 n^{2}-6 n+2}-3 n$ , $a_{n}=\sqrt[n]{2^{n}+e^{n}+\left(\frac{5}{6}\right)^{n}}$ , $a_{n}=\left(\frac{n^{3}+1}{n^{3}}\right)^{5 n^{2}}$ .

See Solution

Problem 31178

Find the derivative of $f(x)=3^{\sec(2x^{3}-4)} \sqrt{\tan^{-1}(x^{3}+1)-2x^{5}}$ .

See Solution

Problem 31179

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}+4 x-5}{x-1}$ . Use l'Hospital's Rule if needed.

See Solution

Problem 31180

Find the curve with positive derivative and length $L=\int_{1}^{4} \sqrt{1+4 x^{2}} d x$ through $(1,0)$ .

See Solution

Problem 31181

Find the limit using l'Hospital's Rule if needed: $\lim _{x \rightarrow(\pi / 2)^{+}} \frac{\cos (x)}{1-\sin (x)}$ and $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{\sin (8 x)}$ .

See Solution

Problem 31182

Calculate the integral: $\int x^{7} \sqrt[3]{2 x^{4}+1} \, dx$ .

See Solution

Problem 31183

Find the $n$ th partial sum formula for each series and determine the sum if it converges.

See Solution

Problem 31184

Evaluate the integral: $\int_{1 / 5}^{e^{2} / 5} \frac{1}{x[\ln (5 x)+3]^{2}} d x$

See Solution

Problem 31185

Find the average rate of change of $f(x) = \frac{4}{3x + 5}$ over the interval $[3, 5]$ .

See Solution

Problem 31186

Find the average rate of change of $f(x)=\frac{2}{2x+3}$ over the interval $[2,4]$ .

See Solution

Problem 31187

Find the relative maximum point of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1$ . Use $f'(x)$ and $f''(x)$ .

See Solution

Problem 31188

Find the limit using l'Hospital's Rule or a simpler method: $\lim _{x \rightarrow \infty} \frac{x+x^{2}}{2-5 x^{2}}$ and $\lim _{x \rightarrow 0} \frac{e^{9 x}-1-9 x}{x^{2}}$ .

See Solution

Problem 31189

Evaluate the integral: $\int \sqrt[3]{(\cos(3 x))^{2}}(\sin(3 x))^{3} d x$

See Solution

Problem 31190

List the first 8 terms and find the sum or divergence for these series:
7. $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{4^{n}}$
8. $\sum_{n=2}^{\infty} \frac{1}{4^{n}}$
9. $\sum_{n=1}^{\infty}\left(1-\frac{7}{4^{n}}\right)$
10. $\sum_{n=0}^{\infty}(-1)^{n} \frac{5}{4^{n}}$
11. $\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right)$
12. $\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}-\frac{1}{3^{n}}\right)$
13. $\sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}+\frac{(-1)^{n}}{5^{n}}\right)$
14. $\sum_{n=0}^{\infty}\left(\frac{2^{n+1}}{5^{n}}\right)$

See Solution

Problem 31191

Determine if the series $1+\left(\frac{2}{5}\right)+\left(\frac{2}{5}\right)^{2}+\cdots$ converges or diverges. Find the sum if it converges.

See Solution

Problem 31192

Find the limit using l'Hospital's rule if needed: $\lim _{x \rightarrow 0} \frac{e^{9 x}-1-9 x}{x^{2}}$ .

See Solution

Problem 31193

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{9 x}-1-9 x}{x^{2}}$ . Use l'Hospital's rule if needed.

See Solution

Problem 31194

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

See Solution

Problem 31195

Find the limit: $\lim _{x \rightarrow 0} \frac{3 x+\sin (x)}{7 x+\cos (x)}$ . Use I'Hospital's Rule if needed.

See Solution

Problem 31196

Find the limit using l'Hospital's Rule if needed: $\lim _{x \rightarrow \infty} \frac{(\ln (x))^{2}}{5 x}$
Also, find this limit: $\lim _{x \rightarrow 0} \frac{3 x+\sin (x)}{7 x+\cos (x)}$

See Solution

Problem 31197

Find the average rate of change of $f(x)=x^{2}+7x$ on $[-3,-3+h]$ . Express your answer in terms of $h$ .

See Solution

Problem 31198

Find the function with derivative $f^{\prime}(x)=e^{7 x}$ that goes through $P=(0,6/7)$ . What is $f(x)=?$

See Solution

Problem 31199

Find the limit: $\lim _{x \rightarrow \infty} x \sin \left(\frac{2 \pi}{x}\right)$ . Use l'Hospital's Rule if needed.

See Solution

Problem 31200

Find the limit: $\lim _{x \rightarrow 0^{+}} \cot (5 x) \sin (15 x)$ . What indeterminate form is it? Options: $0^{0}$ , $\infty^{0}$ , $\frac{\infty}{0}$ , $1^{\infty}$ , $\infty \cdot 0$ .

See Solution

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Calculus

Problem 31101

Find the tangent line approximation for f(3.15)f(3.15)f(3.15) given f(3)=−2f(3)=-2f(3)=−2 and f′(3)=4f'(3)=4f′(3)=4.

Problem 31102

Approximate f(−3.85)f(-3.85)f(−3.85) using the tangent line at x=−4x=-4x=−4 where f(−4)=−8f(-4)=-8f(−4)=−8 and f′(x)=−x−6f^{\prime}(x)=-x-6f′(x)=−x−6.

Problem 31103

Find the tangent line at x=−3x=-3x=−3 for fff with f(−3)=0f(-3)=0f(−3)=0 and f′(x)=−5x−6f^{\prime}(x)=-5x-6f′(x)=−5x−6. Estimate f(−2.85)f(-2.85)f(−2.85).

Problem 31104

Calcule a taxa média de variação de f(x)f(x)f(x) no intervalo 3≤x≤43 \leq x \leq 43≤x≤4 usando os pontos (3,−1)(3,-1)(3,−1) e (4,1)(4,1)(4,1).

Problem 31105

Find the average rate of change of g(x)=−x2+2x+8g(x)=-x^{2}+2 x+8g(x)=−x2+2x+8 from x=−3x=-3x=−3 to x=3x=3x=3.

Problem 31106

The town's population is 20000 and decreases by 1.6%1.6\%1.6% per year. How long until it reaches 15000?

Problem 31107

Solve the ODE dzdx−8xz=8x2−1\frac{d z}{d x}-8 x z=8 x^{2}-1dxdz​−8xz=8x2−1 with initial condition z(1)=e−1z(1)=e-1z(1)=e−1.

Problem 31108

Problem 31109

Find the slope of the tangent line to f−1(x)f^{-1}(x)f−1(x) at (0,1) where f(x)=2−x−x3f(x)=2-x-x^{3}f(x)=2−x−x3. Provide the exact answer.

Problem 31110

Find the average rate of change of horsepower, H(s)=0.003s2+0.07s−0.027H(s)=0.003 s^{2}+0.07 s-0.027H(s)=0.003s2+0.07s−0.027, as speed increases from 80 mph to 100 mph.

Problem 31111

Find the average rate of change of f(x)=2x+5f(x) = 2 \sqrt{x} + 5f(x)=2x​+5 on the interval [4,9][4, 9][4,9].

Problem 31112

Problem 31113

A store's profit function for fans is P(x)=−x2+39x−240P(x)=-x^{2}+39 x-240P(x)=−x2+39x−240. Find the average profit change for price increases from \$10 to \$11 and from \$20 to \$21.

Problem 31114

Find the rate of change of xxx when x=π2x=\frac{\pi}{2}x=2π​, given y=2ecos⁡(x)y=2 e^{\cos (x)}y=2ecos(x) and dydt=5\frac{dy}{dt}=5dtdy​=5.

Problem 31115

Find the expression for f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ where f(x)=4x2−9f(x)=4 x^{2}-9f(x)=4x2−9.

Problem 31116

Evaluate the limit: lim⁡(x,y)→(8,7)x−y−1x−y−1= \lim _{(x, y) \rightarrow(8,7)} \frac{x-y-1}{\sqrt{x-y}-1}= (x,y)→(8,7)lim​x−y​−1x−y−1​=

Problem 31117

Problem 31118

Prove that for all x,y∈Rx, y \in \mathrm{R}x,y∈R, the inequality ∣arctan⁡(x3)−arctan⁡(y3)∣≤M∣x−y∣|\arctan(x^3) - \arctan(y^3)| \leq M|x-y|∣arctan(x3)−arctan(y3)∣≤M∣x−y∣ holds, where M=sup⁡t∈x3t21+t6M = \sup_{t \in x} \frac{3t^2}{1+t^6}M=supt∈x​1+t63t2​.

Problem 31119

Find the limit: lim⁡x→−37x2\lim _{x \rightarrow-3} 7 x^{2}x→−3lim​7x2. Complete the table for values around −3-3−3.

Problem 31120

Find the limit: lim⁡x→1(−7x3)\lim _{x \rightarrow 1}\left(-7 x^{3}\right)x→1lim​(−7x3). Fill in the table for f(x)=−7x3f(x)=-7 x^{3}f(x)=−7x3 at values near 1.

Problem 31121

Find the displacement of a sled starting from rest, accelerating at 50 m/s250 \mathrm{~m/s^2}50 m/s2 for 5 s5 \mathrm{~s}5 s.

Problem 31122

Find the limit: lim⁡x→39x3\lim _{x \rightarrow 3} 9 x^{3}x→3lim​9x3. Fill in the table for f(x)=9x3f(x)=9 x^{3}f(x)=9x3 at values around x=3x=3x=3.

Problem 31123

Find the limit: lim⁡x→3(−3x2)\lim _{x \rightarrow 3}\left(-3 x^{2}\right)x→3lim​(−3x2) by completing the table for f(x)=−3x2f(x)=-3 x^{2}f(x)=−3x2.

Problem 31124

Find the limit: lim⁡x→−2(−4x2)\lim _{x \rightarrow-2}(-4 x^{2})x→−2lim​(−4x2) using a table of values around x=−2x = -2x=−2.

Problem 31125

Find the limit: lim⁡x→3(−3x2)\lim _{x \rightarrow 3}\left(-3 x^{2}\right)x→3lim​(−3x2). Complete the table for f(x)=−3x2f(x)=-3 x^{2}f(x)=−3x2.

Problem 31126

Find the derivative f′(x)f'(x)f′(x) for the function f(x)=−8ln⁡x+5x2−12f(x)=-8 \ln x + 5 x^2 - 12f(x)=−8lnx+5x2−12.

Problem 31127

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=3ex+2x−5ln⁡xf(x)=3 e^{x}+2 x-5 \ln xf(x)=3ex+2x−5lnx.

Problem 31128

Differentiate the function f(x)=−3+2x+2exf(x)=-3+2x+2e^{x}f(x)=−3+2x+2ex. Find f′(x)f^{\prime}(x)f′(x).

Problem 31129

Find the derivative of f(x)=ln⁡x10f(x)=\ln x^{10}f(x)=lnx10 with respect to xxx. What is dfdx\frac{df}{dx}dxdf​?

Problem 31130

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=ln⁡x9+7ln⁡xf(x)=\ln x^{9}+7 \ln xf(x)=lnx9+7lnx.

Problem 31131

Find the limit as xxx approaches -7: lim⁡x→−7x=□\lim _{x \rightarrow-7} x = \squarelimx→−7​x=□ or it does not exist.

Problem 31132

Find the second order Taylor polynomial of fff at x0=0x_0=0x0​=0 given f(0)=0f(0)=0f(0)=0 and x+f(x)+ln⁡(1+f(x))=0x+f(x)+\ln(1+f(x))=0x+f(x)+ln(1+f(x))=0.

Problem 31133

Find the limit: lim⁡x→8(6x−4)\lim _{x \rightarrow 8}(6 x-4)limx→8​(6x−4). Choose A for a value or B if it doesn't exist.

Problem 31134

Find the limit: lim⁡x→−38x4\lim _{x \rightarrow-3} 8 x^{4}limx→−3​8x4. Is it an integer, fraction, or does it not exist?

Problem 31135

Find the limit: lim⁡x→1−5\lim _{x \rightarrow 1}-5limx→1​−5. Choose A or B and fill in the answer.

Problem 31136

Find the limit: lim⁡x→4xx+5\lim _{x \rightarrow 4} \frac{x}{x+5}x→4lim​x+5x​. A. Answer: □\square□ (integer or simplified fraction) B. Limit does not exist.

Problem 31137

Find the limit: lim⁡x→6x2−36x−6\lim _{x \rightarrow 6} \frac{x^{2}-36}{x-6}x→6lim​x−6x2−36​. Choose A (value) or B (does not exist).

Problem 31138

Rewrite f(x)=15x+ln⁡(15)+ln⁡(x)f(x) = 15x + \ln(15) + \ln(x)f(x)=15x+ln(15)+ln(x) using logarithm properties, then find f′(x)f^{\prime}(x)f′(x).

Problem 31139

Given the function f(x)=2cos⁡2(x)−4sin⁡(x)f(x)=2 \cos ^{2}(x)-4 \sin (x)f(x)=2cos2(x)−4sin(x) for 0≤x≤2π0 \leq x \leq 2 \pi0≤x≤2π, find:(a) Intervals of increase and decrease. (b) Local min and max values. (c) Inflection points and intervals of concavity.

Problem 31140

Find the limit as hhh approaches 0 for 9h+1−1h\frac{\sqrt{9 h+1}-1}{h}h9h+1​−1​. Is it A. □\square□ or B. Does not exist?

Problem 31141

Find the limit: lim⁡x→35x−15x−3\lim _{x \rightarrow 3} \frac{5 x-15}{x-3}x→3lim​x−35x−15​. Choose A (value) or B (limit does not exist).

Find the tangent line approximation for $f(3.15)$ given $f(3)=-2$ and $f'(3)=4$ .

Approximate $f(-3.85)$ using the tangent line at $x=-4$ where $f(-4)=-8$ and $f^{\prime}(x)=-x-6$ .

Find the tangent line at $x=-3$ for $f$ with $f(-3)=0$ and $f^{\prime}(x)=-5x-6$ . Estimate $f(-2.85)$ .

Calcule a taxa média de variação de $f(x)$ no intervalo $3 \leq x \leq 4$ usando os pontos $(3,-1)$ e $(4,1)$ .

Find the average rate of change of $g(x)=-x^{2}+2 x+8$ from $x=-3$ to $x=3$ .

The town's population is 20000 and decreases by $1.6\%$ per year. How long until it reaches 15000?

Solve the ODE $\frac{d z}{d x}-8 x z=8 x^{2}-1$ with initial condition $z(1)=e-1$ .

Find the slope of the tangent line to $f^{-1}(x)$ at (0,1) where $f(x)=2-x-x^{3}$ . Provide the exact answer.

Find the average rate of change of horsepower, $H(s)=0.003 s^{2}+0.07 s-0.027$ , as speed increases from 80 mph to 100 mph.

Find the average rate of change of $f(x) = 2 \sqrt{x} + 5$ on the interval $[4, 9]$ .

A store's profit function for fans is $P(x)=-x^{2}+39 x-240$ . Find the average profit change for price increases from \$10 to \$11 and from \$20 to \$21.

Find the rate of change of $x$ when $x=\frac{\pi}{2}$ , given $y=2 e^{\cos (x)}$ and $\frac{dy}{dt}=5$ .

Find the expression for $\frac{f(x+h)-f(x)}{h}$ where $f(x)=4 x^{2}-9$ .

Evaluate the limit:
$\lim _{(x, y) \rightarrow(8,7)} \frac{x-y-1}{\sqrt{x-y}-1}=$

Prove that for all $x, y \in \mathrm{R}$ , the inequality $|\arctan(x^3) - \arctan(y^3)| \leq M|x-y|$ holds, where $M = \sup_{t \in x} \frac{3t^2}{1+t^6}$ .

Find the limit: $\lim _{x \rightarrow-3} 7 x^{2}$ . Complete the table for values around $-3$ .

Find the limit: $\lim _{x \rightarrow 1}\left(-7 x^{3}\right)$ . Fill in the table for $f(x)=-7 x^{3}$ at values near 1.

Find the displacement of a sled starting from rest, accelerating at $50 \mathrm{~m/s^2}$ for $5 \mathrm{~s}$ .

Find the limit: $\lim _{x \rightarrow 3} 9 x^{3}$ . Fill in the table for $f(x)=9 x^{3}$ at values around $x=3$ .

Find the limit: $\lim _{x \rightarrow 3}\left(-3 x^{2}\right)$ by completing the table for $f(x)=-3 x^{2}$ .

Find the limit: $\lim _{x \rightarrow-2}(-4 x^{2})$ using a table of values around $x = -2$ .

Find the limit: $\lim _{x \rightarrow 3}\left(-3 x^{2}\right)$ . Complete the table for $f(x)=-3 x^{2}$ .

Find the derivative $f'(x)$ for the function $f(x)=-8 \ln x + 5 x^2 - 12$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=3 e^{x}+2 x-5 \ln x$ .

Differentiate the function $f(x)=-3+2x+2e^{x}$ . Find $f^{\prime}(x)$ .

Find the derivative of $f(x)=\ln x^{10}$ with respect to $x$ . What is $\frac{df}{dx}$ ?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\ln x^{9}+7 \ln x$ .

Find the limit as $x$ approaches -7: $\lim _{x \rightarrow-7} x = \square$ or it does not exist.

Find the second order Taylor polynomial of $f$ at $x_0=0$ given $f(0)=0$ and $x+f(x)+\ln(1+f(x))=0$ .

Find the limit: $\lim _{x \rightarrow 8}(6 x-4)$ . Choose A for a value or B if it doesn't exist.

Find the limit: $\lim _{x \rightarrow-3} 8 x^{4}$ . Is it an integer, fraction, or does it not exist?

Find the limit: $\lim _{x \rightarrow 1}-5$ . Choose A or B and fill in the answer.

Find the limit: $\lim _{x \rightarrow 4} \frac{x}{x+5}$ . A. Answer: $\square$ (integer or simplified fraction) B. Limit does not exist.

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{2}-36}{x-6}$ . Choose A (value) or B (does not exist).

Rewrite $f(x) = 15x + \ln(15) + \ln(x)$ using logarithm properties, then find $f^{\prime}(x)$ .

Given the function $f(x)=2 \cos ^{2}(x)-4 \sin (x)$ for $0 \leq x \leq 2 \pi$ , find:
(a) Intervals of increase and decrease. (b) Local min and max values. (c) Inflection points and intervals of concavity.

Find the limit as $h$ approaches 0 for $\frac{\sqrt{9 h+1}-1}{h}$ . Is it A. $\square$ or B. Does not exist?

Find the limit: $\lim _{x \rightarrow 3} \frac{5 x-15}{x-3}$ . Choose A (value) or B (limit does not exist).

Find $f^{\prime}(x)$ for $f(x)=-6x+4$ and determine the tangent line slopes at $x=-4$ and $x=4$ .

Find the derivative of $f(x)=x^{2}-4x+8$ and the slope at $x=\frac{3}{2}$ and $x=2$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin x}$ .

Find $\frac{d f^{-1}}{d x}$ at $x=f(3)$ if $\frac{d f(x)}{d x}=x^{2}-3 x+2$ for $x>1$ . Choices: a. 2 b. 3 c. none d. $\frac{1}{2}$ e. $\frac{1}{3}$

Find the slope and equation of the tangent line for $f(x)=2x^{2}-5x$ at the point (2,-2). Slope: $\square$ .

Find the limit: $\lim _{x \rightarrow 2}\left(3 x^{2}-8 x+7\right)^{2}$ . Is it A. $\square$ or B. does not exist?

Rewrite $f(x)=7 \ln \left(\frac{14}{x}\right)$ using logarithm properties, then find $f^{\prime}(x)$ . $f^{\prime}(x)=\square$

Find the derivative of $y=8 \log _{7} x$ . What is $\frac{d y}{d x}=\square$ ?

Find the limit: $\lim _{x \rightarrow 0^{+}}\left[\frac{1}{\operatorname{Arctan} x}-\frac{1}{x}\right]$ .

Evaluate $\lim_{x \rightarrow \infty} f(x)$ for $f(x)=\frac{2x^{2}+3x}{(x+1)(x+2)}$ to find horizontal asymptotes.

Find the derivative $\frac{d y}{d x}$ for the function $y=5^{x}$ . What is $\frac{d y}{d x}=\square$ ?

Find the derivative $\frac{d y}{d x}$ for the function $y=6^{x}+e^{4}$ . Type an exact answer.

Find the annual depreciation rate for the function $S(t)=700,000(0.8)^{t}$ after 1 and 5 years.

Model train position is given by $s(t)=2.5 t+17$ . Find speed, position after 4 sec, and time to reach 32 feet.

Find the limit: $\lim _{\theta \rightarrow 0^{+}}\left[\csc \theta-\frac{\cos \sqrt{\theta}}{\sin \theta}\right]$ .

Find (A) the derivative of $F(x) S(x)$ without the product rule, and (B) $F^{\prime}(x) S^{\prime}(x)$ . Given $F(x)=x^{4}+1$ , $S(x)=x^{5}$ .

Finde die Kostenfunktion $K$ aus den Ableitungen $K^{\prime}(x)$ und beschreibe den Kostenverlauf.

Find $f(x+h) - f(x)$ and $\frac{f(x+h) - f(x)}{h}$ for the function $f(x) = 6 - x^{2}$ .

Evaluate the limits: $-x^{2} \leq \lim _{x \rightarrow 0} x^{2} \sin \left(\frac{1}{x}\right) \leq x^{2}$ .

A bucket with a mass of $2.30 \mathrm{~kg}$ is whirled in a circle of radius $1.40 \mathrm{~m}$ . Find the acceleration at the top ( $\mathrm{m} / \mathrm{s}^2$ ).

Find $\lim _{\theta \rightarrow 0} \frac{\sin (\theta)}{\theta}$ using the inequality $\cos (\theta) \leq \frac{\sin (\theta)}{\theta} \leq 1$ .

Find the limit: $\lim _{x \rightarrow+\infty}\left(1-\frac{5}{x}\right)^{3 x}$ .

Find the integral of $(\ln (x))^{2}$ with respect to $x$ .

Find the integral of the function $y e^{-y}$ with respect to $y$ : $\int y e^{-y} d y$ .

Calculate the integral: $\int(\pi-x) \cos (\pi x) \, dx$ .

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}}\left[\ln \left(\frac{1}{1-x}\right)\right]^{1-x}$ .

Find the slope and tangent line equation at point $P(4,36)$ for the curve $y=x^{2}+5x$ .