Calculus

Problem 33101

Find the limit: $\lim _{x \rightarrow+\infty} \frac{7-6 x^{5}}{x+3}$ .

See Solution

Problem 33102

Find $\frac{d y}{d t}$ at $x=5$ for $y=x^{2}+5$ given $\frac{d x}{d t}=30$ .

See Solution

Problem 33103

Find the limit as $t$ approaches infinity for the expression $\frac{6-t^{3}}{7 t^{3}+3}$ .

See Solution

Problem 33104

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+10}-\sqrt{x^{2}-10}\right)$ .

See Solution

Problem 33105

Find the area under the curve from -5 to 5 for $5 - |x|$ . Compute the integral: $\int_{-5}^{5}(5-|x|) d x$

See Solution

Problem 33106

Find the limit: $\lim _{x \rightarrow+\infty} \sqrt[3]{\frac{2+3 x-5 x^{2}}{1+8 x^{2}}}$ .

See Solution

Problem 33107

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{5 x^{2}-2}}{x+3}$ .

See Solution

Problem 33108

A student kicks a football to a max height of $50.0 \mathrm{~m}$ . Find the kick speed and total flight time.

See Solution

Problem 33109

Find the horizontal asymptotes of $f(x)=\frac{\sqrt{7 x^{2}+8}}{9 x+7}$ . Provide answers as a comma-separated list.

See Solution

Problem 33110

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{8 x^{4}+7 x+7}}{7 x^{3}+1}=\square$

See Solution

Problem 33111

Find all numbers $x$ in $(-1,3)$ where $f'(x)=3x^2+6x-1$ equals the average rate of change of $f(x)=x^3+3x^2-x+7$ .

See Solution

Problem 33112

Find the tangent line equation at (2,-10) for $f(x)=6-2x^3$ using the limit definition of the derivative: $y=mx+b$ .

See Solution

Problem 33113

Given the piecewise function
$f(x)=\left\{\begin{array}{ll} 2 x^{2}+5, & x<0 \\ \frac{3-5 x^{3}}{1+4 x+x^{3}}, & x \geq 0 \end{array}\right.$
find: a. $\lim _{x \rightarrow-\infty} f(x)$ b. $\lim _{x \rightarrow+\infty} f(x)$ .

See Solution

Problem 33114

Find the truncated Taylor series for $\ln(y)$ at $y=1$ for $n=0$ , $n=1$ , and $n=2$ . Compare errors at $y=1.1$ and $y=1.75$ .

See Solution

Problem 33115

An artillery shell is fired at $29.9^{\circ}$ with a speed of $1560 \mathrm{~m/s}$ . Find the flight time in minutes, ignoring air resistance.

See Solution

Problem 33116

Complete the table and estimate the limit: $f(x)=\frac{\sqrt{x^{2}+x}}{x+1} ; \quad \lim _{x \rightarrow+\infty} f(x)$

See Solution

Problem 33117

Calculate the average rate of change of $R(t)$ from $t=2$ to $t=6$ where $R(2)=20.7$ , $R(6)=19.9$ .

See Solution

Problem 33118

Define the piecewise function $f(x)$ and find where it is discontinuous and not differentiable. Sketch the graph.

See Solution

Problem 33119

Evaluate the limits for $f(x)=|x-2|$ :
1. $\lim _{x \rightarrow 2^{-}} \frac{f(x)-f(2)}{x-2}=\square$
2. $\lim _{x \rightarrow 2^{+}} \frac{f(x)-f(2)}{x-2}=\square$
Is $f(x)$ differentiable at 2?

See Solution

Problem 33120

Find the limit as $t$ approaches 0 for the expression $\frac{\cot t}{\csc t}$ .

See Solution

Problem 33121

Find the total oil leaked in the first 10 hours after the tanker breaks apart, given $R(t)=\frac{0.6}{1+t^{2}}$ .

See Solution

Problem 33122

Evaluate the integral: $\int \frac{78}{\sqrt[3]{x}} dx = C$

See Solution

Problem 33123

Given $f(x)=\frac{4}{x}$ , find $A$ in $\frac{f(x+h)-f(x)}{h}=\frac{A}{x(x+h)}$ . Then compute $f^{\prime}(1)$ , $f^{\prime}(2)$ , $f^{\prime}(3)$ .

See Solution

Problem 33124

Indicate T or F for each statement below. All must be correct for credit.
1. If $\lim _{x \rightarrow 7}[f(x) g(x)]$ exists, then it's $f(7) g(7)$
2. If $\lim _{x \rightarrow 6} f(x)=\infty$ and $\lim _{x \rightarrow 6} g(x)=\infty$ , then $\lim _{x \rightarrow 6}[f(x)-g(x)]=0$
3. $\lim _{x \rightarrow 3} \frac{x^{2}+6 x-27}{x^{2}+7 x-30}=\frac{\lim _{x \rightarrow 3} x^{2}+6 x-27}{\lim _{x \rightarrow 3} x^{2}+7 x-30}$
4. If $f(x)$ is differentiable at $a$ , then $f(x)$ is continuous at $a$
5. If $\lim _{x \rightarrow 6} f(x)=6$ and $\lim _{x \rightarrow 6} g(x)=0$ , then $\lim _{x \rightarrow 6}[f(x) / g(x)]$ does not exist

See Solution

Problem 33125

Indicate True (T) or False (F) for these statements about limits and continuity:
1. If $\lim _{x \rightarrow f}[f(x) g(x)]$ exists, then it equals $f(7) g(7)$ .
2. If $\lim _{x \rightarrow 6} f(x)=\infty$ and $\lim _{x \rightarrow \infty} g(x)=\infty$ , then $\lim _{x \rightarrow 6}[f(x)-g(x)]=0$ .
3. $\lim _{x \rightarrow 3} \frac{x^{2}+6 x-27}{x^{2}+7 x-30}=\frac{\lim _{x \rightarrow 3} (x^{2}+6 x-27)}{\lim _{x \rightarrow 3} (x^{2}+7 x-30)}$ .
4. If $f(x)$ is differentiable at $a$ , then $f(x)$ is continuous at $a$ .
5. If $\lim _{x \rightarrow 6} f(x)=6$ and $\lim _{x \rightarrow 6} g(x)=0$ , then $\lim _{x \rightarrow 6}[f(x) / g(x)]$ does not exist.

See Solution

Problem 33126

Determine if the sequence $a_{n}=\frac{2 n^{2}+n-1}{n^{2}}$ converges or diverges, and find the limit if it converges.

See Solution

Problem 33127

Determine the behavior of the curve $x^{2} y - 4 x = 5$ at the point $(5,1)$ : increasing/concave up/down or decreasing/concave up/down?

See Solution

Problem 33128

Find the limit: $\lim _{n \rightarrow \infty} \frac{n^{2}}{\sqrt{n^{3}+4 n}}$ .

See Solution

Problem 33129

Find the derivative of $f(x)=\frac{5}{5x+6}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 33130

Determine if the sequence $a_{n}=\frac{n^{2}}{\sqrt{n^{3}+4 n}}$ converges and find its limit.

See Solution

Problem 33131

Find a relative minimum for the function $h(t)=t-4 \sqrt{1+t}$ using the second derivative test.

See Solution

Problem 33132

Find the derivative of $f(x)=x^{4} \cos x$ . What is $f^{\prime}(x)=?$

See Solution

Problem 33133

Find the critical value of $f(x)=2x+5\cos(x)$ on $0 \leq x \leq 2\pi$ that has absolute and relative extrema. Options: 0, 0.412, 2.730, 6.283.

See Solution

Problem 33134

Find the derivative of the function $f(x)=\left(7 x-5 x^{3}\right)(6+\sqrt{x})$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 33135

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{1}{32+h}-0.03125}{h}$ .

See Solution

Problem 33136

Find the derivative of the function $f(x)=(7x-5x^{3})(6+\sqrt{x})$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 33137

Find $g'(1)$ for $g(x)=2-3x^3$ and use it to write the tangent line at $(1,-1)$ in $y=mx+b$ form.

See Solution

Problem 33138

Find $g(1)$ for $g(x)=2-3x^3$ and use it to find the tangent line at $(1,-1)$ in the form $y=mx+b$ .

See Solution

Problem 33139

Determine if the sequence $a_n = 1 - (0.2)^n$ converges and find its limit.

See Solution

Problem 33140

Deposit \ $1000 at$ 3\%$ simple interest. Find the average rate of change for the first year and six years, to two decimal places.

See Solution

Problem 33141

Find the derivative of the function $f(x)=9 x^{7/5}-5 x^{2}+10^{4}$ .

See Solution

Problem 33142

Find the derivative of the function $f(x)=\cos x-7 \tan x$ , i.e., compute $f^{\prime}(x)$ .

See Solution

Problem 33143

Evaluate the integral: $\int \frac{68 x^{3}+7 x-46}{x^{2}} d x =$

See Solution

Problem 33144

Find the first four terms of the Taylor series for $f(y)=r y \ln \left(\frac{k}{y}\right)$ about $y=K$ .

See Solution

Problem 33145

Find the derivative $f'(x)$ of the function $f(x)=\frac{2-x^{2}}{3+x^{2}}$ and evaluate $f'(3)$ .

See Solution

Problem 33146

Find the derivative $f'(x)$ of the function $f(x)=\frac{\sqrt{x}-2}{\sqrt{x}+2}$ and evaluate $f'(2)$ .

See Solution

Problem 33147

Find the function $f(x)$ and the number $a$ for the limit $\lim _{h \rightarrow 0} \frac{\sqrt{64+h}-8}{h}$ .

See Solution

Problem 33148

Find the limit $\lim _{h \rightarrow 0} \frac{(8+h)^{2}-64}{h}$ and identify the function $f(t)$ and the value $a$ .

See Solution

Problem 33149

Evaluate the integral $\int_{3}^{6} \left(\frac{d}{d t} \sqrt{5+3 t^{4}}\right) d t$ .

See Solution

Problem 33150

1. What is the expected temperature at 10 km if sea level is 22°C and 2 km is 9°C?
2. Find a function $T=f(h)$ for 27°C at sea level. What is its reasonable domain?
3. For pressure $p$ at height $h$ : $p=101.32 \cdot e^{\left(-\frac{34.2 h}{T+273.15}\right)}$ . Find $p$ at 10 km with 27°C.

See Solution

Problem 33151

Find the derivative $y^{\prime}$ of the function $y=\int_{1}^{\cos (x)}(2 t+10 \sin (t)) d t$ using the Fundamental Theorem of Calculus.

See Solution

Problem 33152

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+7 t+2)(6 t^{-2}+6 t^{-3})$ .

See Solution

Problem 33153

Find the derivative of the function $F(x)=e^{x^{2}}$ . What is $F^{\prime}(x)$ ? $F^{\prime}(x)=\square$

See Solution

Problem 33154

Evaluate the integral $\int_{0}^{4} 2 x e^{x^{2}} d x=\square$ using the Fundamental Theorem of Calculus.

See Solution

Problem 33155

Find the function $f(a)$ and the number $a$ such that $\lim _{h \rightarrow 0} \frac{(8+h)^{2}-64}{h}$ represents its derivative.

See Solution

Problem 33156

Find $f(x)$ where $f'(x)=\frac{\tan(x)}{x}$ and $f(1)=5$ .

See Solution

Problem 33157

If $f(3)=14$ , $f^{\prime}$ is continuous, and $\int_{3}^{7} f^{\prime}(t) dt=20$ , find $f(7)$ .

See Solution

Problem 33158

Find and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{5}{x}$ , then use it to find the derivative $f'(z)$ .

See Solution

Problem 33159

Find and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{5}{x}$ , then use it to find the derivative. Also, find $f^{\prime}(2)$ .

See Solution

Problem 33160

Find the open intervals where the function $f(x)=\cos (x)-\sin (x)$ is increasing on $[0,2 \pi]$ .

See Solution

Problem 33161

Given $f(x)=11 x(\sin (x)+\cos (x))$ , find $f^{\prime}(x)$ and $f^{\prime}\left(\frac{\pi}{4}\right)$ .

See Solution

Problem 33162

Evaluate the integral $\int_{-\pi}^{\pi} f(x) \, dx$ for the piecewise function $f(x) = \{-4x \text{ if } -\pi \leq x \leq 0, \sin(x) \text{ if } 0 < x \leq \pi\}$ .

See Solution

Problem 33163

A graphing calculator is recommended.
1. Bismuth-210 has a half-life of 5 days. (a) Find remaining mass after 15 days from 160 mg. (b) Determine remaining mass after $t$ days: $y(t) = \mathrm{mg}$ . (c) Estimate remaining mass after 2 weeks (round to 1 decimal). (d) Find days to reduce mass to 1 mg (round to 1 decimal).

See Solution

Problem 33164

Find the domain of $y=\frac{1}{x^{2}-3}$ and the largest interval $I$ for the initial condition.

See Solution

Problem 33165

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{-55}^{-5} \frac{d x}{x}=$

See Solution

Problem 33166

Find a function with second derivative $y''=12x-2$ and tangent line $y=-x+5$ at $x=1$ . What is $y=?$

See Solution

Problem 33167

Check if these functions are eigenfunctions of the operator $\frac{d}{d x}$ : a. $e^{-i k x}$ , b. $k x^{2}$ .

See Solution

Problem 33168

Find the value of $\int_{4}^{-4}(f(x)+4) d x$ given $\int_{4}^{8} f(x) d x=7$ and $\int_{-4}^{8} f(x) d x=15$ .

See Solution

Problem 33169

Given $y=5 x^{2}$ , find $\Delta y$ when $x=5$ and $\Delta x=0.1$ . Also, find $d y$ for $d x=0.1$ .

See Solution

Problem 33170

Determine the formula for the integral from 4 to $x^{2}$ of $(t+5)$ with respect to $t$ .

See Solution

Problem 33171

Find the function formula for the integral: $\int_{-2}^{x}(8 t^{2}-7 t) dt =$

See Solution

Problem 33172

Find the derivative $y^{\prime}$ of $y=\int_{1}^{\cos (x)}(-10 t+3 \sin (t)) d t$ using the Fundamental Theorem of Calculus.

See Solution

Problem 33173

Find $d y$ for $y=\tan(3x+3)$ at $x=2$ with $d x=0.1$ and $d x=0.2$ .

See Solution

Problem 33174

Find $F(-9)$ , $F(9)$ , $F^{\prime}(0)$ , and $F^{\prime}(9)$ for $F(x)=\int_{-9}^{x} \frac{d u}{u^{2}+1}$ .

See Solution

Problem 33175

Estimate $\Delta y$ for $y=4 x^{2}+6 x+2$ with $\Delta x=0.4$ at $x=3$ using linear approximation.

See Solution

Problem 33176

Given $f(t)$ as the weight of a solid in water, with $f^{\prime}(t)=-1 f(t)(4+f(t))$ . If $f(2)=4$ , find $f(2+\frac{1}{60})$ .

See Solution

Problem 33177

Approximate $\sqrt[3]{125.3}$ using the tangent line of $f(x)=\sqrt[3]{x}$ at $x=125$ . Find $m$ and $b$ for $y=m x+b$ .

See Solution

Problem 33178

Estimate $\Delta y$ for $y=\sin(3x)$ at $x=0$ with $\Delta x=0.3$ using linear approximation. Find percentage error.

See Solution

Problem 33179

Evaluate the integrals for the piecewise function $f(x)=\left\{\begin{array}{ll}2 x, & x \leq 1 \\ 2, & x>1\end{array}\right.$ : a. $\int_{0}^{1} f(x) d x$ , b. $\int_{-1}^{1} f(x) d x$ , c. $\int_{0}^{10} f(x) d x$ , d. $\int_{1/2}^{5} f(x) d x$ .

See Solution

Problem 33180

Approximate $\frac{1}{1.003}$ using linear approximation with $f(x)=\frac{1}{x}$ at a point near 1.003.

See Solution

Problem 33181

Find $F^{\prime}(x)$ for $F(x)=\int_{x}^{-3}-1 \cos(t^{2}) dt$ using the Fundamental Theorem of Calculus.

See Solution

Problem 33182

Find $f(8)$ given that $f(3)=16$ , $f^{\prime}$ is continuous, and $\int_{3}^{8} f^{\prime}(t) dt=20$ .

See Solution

Problem 33183

Estimate the maximum error in the surface area of a sphere with circumference $90 \mathrm{~cm}$ and error $0.5 \mathrm{~cm}$ .

See Solution

Problem 33184

Find the derivative of the function $F(x)=e^{x^{2}}$ . What is $F^{\prime}(x)$ ? $F^{\prime}(x)=\square$

See Solution

Problem 33185

Find the tangent lines to $f(x)=x^{3}+2$ that are parallel to $3x-y-4=0$ . Give both $y$ -intercepts.

See Solution

Problem 33186

Sketch the area between $y=\frac{x^{3}}{25}$ and the x-axis over $[-6,-5]$ . Find the area. Area $=$

See Solution

Problem 33187

Find the tangent lines to $f(x)=x^{3}+2$ that are parallel to the line $3x - y - 4 = 0$ .

See Solution

Problem 33188

Find the derivative of the function $F(x)=e^{x^{2}}$ , i.e., compute $F^{\prime}(x)$ .

See Solution

Problem 33189

Find the first derivative $\frac{d y}{d x}$ at the point $(1,1)$ for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

See Solution

Problem 33190

Evaluate $f(x)=-2x^{2}-4x+5$ at $x=-6$ and $x=4$ . Does Rolle's Theorem apply? Find $c$ where $f'(c)=0$ or enter "DNE".

See Solution

Problem 33191

Evaluate the integral $\int_{3}^{6} \left(\frac{d}{d t} \sqrt{5+5 t^{4}}\right) d t$ using the Fundamental Theorem of Calculus.

See Solution

Problem 33192

Evaluate the integral $\int_{-\pi}^{\pi} f(x) d x$ where $f(x)=\begin{cases}7x & \text{if } -\pi \leq x \leq 0 \\ 10 \sin(x) & \text{if } 0 < x \leq \pi\end{cases}$ . If it doesn't exist, answer "DNE".

See Solution

Problem 33193

Find $\frac{d f^{-1}}{d x}$ for $f(x)=x^{3}-2$ at $x=6$ where $f(2)=6$ .

See Solution

Problem 33194

Evaluate $f(x)=\cos(5\pi x)$ at $x=\frac{4}{5}$ and $x=\frac{6}{5}$ . Does Rolle's Theorem apply? Find $c$ where $f'(c)=0$ .

See Solution

Problem 33195

Find the derivative of $x^{9/2}$ .

See Solution

Problem 33196

Find the tangent line equation to the curve $f(x)=e^{2 x}$ at the point $(0,1)$ .

See Solution

Problem 33197

Find the slope of the tangent line to the curve $y=x e^{6 x}-e^{2 x}$ at $x=0$ . Options: $-1$ , $-2$ , None.

See Solution

Problem 33198

Apply Rolle's Theorem to $f(x)=2 x^{2}-16 x-4$ on $[2,6]$ . How many $c$ values satisfy $f^{\prime}(c)=0$ ? What is $c$ ? Also, check conditions for $[-1,13]$ .

See Solution

Problem 33199

Find the derivative of $y=4 x^{3}-5 x$ and identify stationary points. Provide answers as specified.

See Solution

Problem 33200

Calculate the integral $\int_{2}^{4} x^{2 x}(1+\ln x) d x$ .

See Solution

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Calculus

Problem 33101

Find the limit: lim⁡x→+∞7−6x5x+3\lim _{x \rightarrow+\infty} \frac{7-6 x^{5}}{x+3}limx→+∞​x+37−6x5​.

Problem 33102

Find dydt\frac{d y}{d t}dtdy​ at x=5x=5x=5 for y=x2+5y=x^{2}+5y=x2+5 given dxdt=30\frac{d x}{d t}=30dtdx​=30.

Problem 33103

Find the limit as ttt approaches infinity for the expression 6−t37t3+3\frac{6-t^{3}}{7 t^{3}+3}7t3+36−t3​.

Problem 33104

Find the limit: lim⁡x→∞(x2+10−x2−10)\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+10}-\sqrt{x^{2}-10}\right)x→∞lim​(x2+10​−x2−10​).

Problem 33105

Find the area under the curve from -5 to 5 for 5−∣x∣5 - |x|5−∣x∣. Compute the integral: ∫−55(5−∣x∣)dx\int_{-5}^{5}(5-|x|) d x∫−55​(5−∣x∣)dx

Problem 33106

Find the limit: lim⁡x→+∞2+3x−5x21+8x23\lim _{x \rightarrow+\infty} \sqrt[3]{\frac{2+3 x-5 x^{2}}{1+8 x^{2}}}limx→+∞​31+8x22+3x−5x2​​.

Problem 33107

Find the limit: lim⁡x→−∞5x2−2x+3\lim _{x \rightarrow-\infty} \frac{\sqrt{5 x^{2}-2}}{x+3}limx→−∞​x+35x2−2​​.

Problem 33108

A student kicks a football to a max height of 50.0 m50.0 \mathrm{~m}50.0 m. Find the kick speed and total flight time.

Problem 33109

Find the horizontal asymptotes of f(x)=7x2+89x+7f(x)=\frac{\sqrt{7 x^{2}+8}}{9 x+7}f(x)=9x+77x2+8​​. Provide answers as a comma-separated list.

Problem 33110

Find the limit: lim⁡x→∞8x4+7x+77x3+1=□\lim _{x \rightarrow \infty} \frac{\sqrt{8 x^{4}+7 x+7}}{7 x^{3}+1}=\squarex→∞lim​7x3+18x4+7x+7​​=□

Problem 33111

Find all numbers xxx in (−1,3)(-1,3)(−1,3) where f′(x)=3x2+6x−1f'(x)=3x^2+6x-1f′(x)=3x2+6x−1 equals the average rate of change of f(x)=x3+3x2−x+7f(x)=x^3+3x^2-x+7f(x)=x3+3x2−x+7.

Problem 33112

Find the tangent line equation at (2,-10) for f(x)=6−2x3f(x)=6-2x^3f(x)=6−2x3 using the limit definition of the derivative: y=mx+by=mx+by=mx+b.

Problem 33113

Problem 33114

Find the truncated Taylor series for ln⁡(y)\ln(y)ln(y) at y=1y=1y=1 for n=0n=0n=0, n=1n=1n=1, and n=2n=2n=2. Compare errors at y=1.1y=1.1y=1.1 and y=1.75y=1.75y=1.75.

Problem 33115

An artillery shell is fired at 29.9∘29.9^{\circ}29.9∘ with a speed of 1560 m/s1560 \mathrm{~m/s}1560 m/s. Find the flight time in minutes, ignoring air resistance.

Problem 33116

Complete the table and estimate the limit: f(x)=x2+xx+1;lim⁡x→+∞f(x)f(x)=\frac{\sqrt{x^{2}+x}}{x+1} ; \quad \lim _{x \rightarrow+\infty} f(x)f(x)=x+1x2+x​​;x→+∞lim​f(x)

Problem 33117

Calculate the average rate of change of R(t)R(t)R(t) from t=2t=2t=2 to t=6t=6t=6 where R(2)=20.7R(2)=20.7R(2)=20.7, R(6)=19.9R(6)=19.9R(6)=19.9.

Problem 33118

Define the piecewise function f(x)f(x)f(x) and find where it is discontinuous and not differentiable. Sketch the graph.

Problem 33119

Problem 33120

Find the limit as ttt approaches 0 for the expression cot⁡tcsc⁡t\frac{\cot t}{\csc t}csctcott​.

Problem 33121

Find the total oil leaked in the first 10 hours after the tanker breaks apart, given R(t)=0.61+t2R(t)=\frac{0.6}{1+t^{2}}R(t)=1+t20.6​.

Problem 33122

Evaluate the integral: ∫78x3dx=C\int \frac{78}{\sqrt[3]{x}} dx = C∫3x​78​dx=C

Problem 33123

Given f(x)=4xf(x)=\frac{4}{x}f(x)=x4​, find AAA in f(x+h)−f(x)h=Ax(x+h)\frac{f(x+h)-f(x)}{h}=\frac{A}{x(x+h)}hf(x+h)−f(x)​=x(x+h)A​. Then compute f′(1)f^{\prime}(1)f′(1), f′(2)f^{\prime}(2)f′(2), f′(3)f^{\prime}(3)f′(3).

Problem 33124

Problem 33125

Problem 33126

Determine if the sequence an=2n2+n−1n2a_{n}=\frac{2 n^{2}+n-1}{n^{2}}an​=n22n2+n−1​ converges or diverges, and find the limit if it converges.

Problem 33127

Determine the behavior of the curve x2y−4x=5x^{2} y - 4 x = 5x2y−4x=5 at the point (5,1)(5,1)(5,1): increasing/concave up/down or decreasing/concave up/down?

Problem 33128

Find the limit: lim⁡n→∞n2n3+4n\lim _{n \rightarrow \infty} \frac{n^{2}}{\sqrt{n^{3}+4 n}}limn→∞​n3+4n​n2​.

Problem 33129

Find the derivative of f(x)=55x+6f(x)=\frac{5}{5x+6}f(x)=5x+65​. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 33130

Determine if the sequence an=n2n3+4na_{n}=\frac{n^{2}}{\sqrt{n^{3}+4 n}}an​=n3+4n​n2​ converges and find its limit.

Problem 33131

Find a relative minimum for the function h(t)=t−41+th(t)=t-4 \sqrt{1+t}h(t)=t−41+t​ using the second derivative test.

Problem 33132

Find the derivative of f(x)=x4cos⁡xf(x)=x^{4} \cos xf(x)=x4cosx. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 33133

Find the critical value of f(x)=2x+5cos⁡(x)f(x)=2x+5\cos(x)f(x)=2x+5cos(x) on 0≤x≤2π0 \leq x \leq 2\pi0≤x≤2π that has absolute and relative extrema. Options: 0, 0.412, 2.730, 6.283.

Problem 33134

Find the derivative of the function f(x)=(7x−5x3)(6+x)f(x)=\left(7 x-5 x^{3}\right)(6+\sqrt{x})f(x)=(7x−5x3)(6+x​). What is f′(x)f^{\prime}(x)f′(x)?

Problem 33135

Find the limit: lim⁡h→0132+h−0.03125h\lim _{h \rightarrow 0} \frac{\frac{1}{32+h}-0.03125}{h}limh→0​h32+h1​−0.03125​.

Problem 33136

Find the derivative of the function f(x)=(7x−5x3)(6+x)f(x)=(7x-5x^{3})(6+\sqrt{x})f(x)=(7x−5x3)(6+x​). What is f′(x)f^{\prime}(x)f′(x)?

Problem 33137

Find g′(1)g'(1)g′(1) for g(x)=2−3x3g(x)=2-3x^3g(x)=2−3x3 and use it to write the tangent line at (1,−1)(1,-1)(1,−1) in y=mx+by=mx+by=mx+b form.

Problem 33138

Find g(1)g(1)g(1) for g(x)=2−3x3g(x)=2-3x^3g(x)=2−3x3 and use it to find the tangent line at (1,−1)(1,-1)(1,−1) in the form y=mx+by=mx+by=mx+b.

Problem 33139

Determine if the sequence an=1−(0.2)na_n = 1 - (0.2)^nan​=1−(0.2)n converges and find its limit.

Problem 33140

Deposit \1000at1000 at 1000at3\%$ simple interest. Find the average rate of change for the first year and six years, to two decimal places.

Problem 33141

Find the derivative of the function f(x)=9x7/5−5x2+104f(x)=9 x^{7/5}-5 x^{2}+10^{4}f(x)=9x7/5−5x2+104.

Problem 33142

Find the limit: $\lim _{x \rightarrow+\infty} \frac{7-6 x^{5}}{x+3}$ .

Find $\frac{d y}{d t}$ at $x=5$ for $y=x^{2}+5$ given $\frac{d x}{d t}=30$ .

Find the limit as $t$ approaches infinity for the expression $\frac{6-t^{3}}{7 t^{3}+3}$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+10}-\sqrt{x^{2}-10}\right)$ .

Find the area under the curve from -5 to 5 for $5 - |x|$ . Compute the integral: $\int_{-5}^{5}(5-|x|) d x$

Find the limit: $\lim _{x \rightarrow+\infty} \sqrt[3]{\frac{2+3 x-5 x^{2}}{1+8 x^{2}}}$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{5 x^{2}-2}}{x+3}$ .

A student kicks a football to a max height of $50.0 \mathrm{~m}$ . Find the kick speed and total flight time.

Find the horizontal asymptotes of $f(x)=\frac{\sqrt{7 x^{2}+8}}{9 x+7}$ . Provide answers as a comma-separated list.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{8 x^{4}+7 x+7}}{7 x^{3}+1}=\square$

Find all numbers $x$ in $(-1,3)$ where $f'(x)=3x^2+6x-1$ equals the average rate of change of $f(x)=x^3+3x^2-x+7$ .

Find the tangent line equation at (2,-10) for $f(x)=6-2x^3$ using the limit definition of the derivative: $y=mx+b$ .

Find the truncated Taylor series for $\ln(y)$ at $y=1$ for $n=0$ , $n=1$ , and $n=2$ . Compare errors at $y=1.1$ and $y=1.75$ .

An artillery shell is fired at $29.9^{\circ}$ with a speed of $1560 \mathrm{~m/s}$ . Find the flight time in minutes, ignoring air resistance.

Complete the table and estimate the limit: $f(x)=\frac{\sqrt{x^{2}+x}}{x+1} ; \quad \lim _{x \rightarrow+\infty} f(x)$

Calculate the average rate of change of $R(t)$ from $t=2$ to $t=6$ where $R(2)=20.7$ , $R(6)=19.9$ .

Define the piecewise function $f(x)$ and find where it is discontinuous and not differentiable. Sketch the graph.

Find the limit as $t$ approaches 0 for the expression $\frac{\cot t}{\csc t}$ .

Find the total oil leaked in the first 10 hours after the tanker breaks apart, given $R(t)=\frac{0.6}{1+t^{2}}$ .

Evaluate the integral: $\int \frac{78}{\sqrt[3]{x}} dx = C$

Given $f(x)=\frac{4}{x}$ , find $A$ in $\frac{f(x+h)-f(x)}{h}=\frac{A}{x(x+h)}$ . Then compute $f^{\prime}(1)$ , $f^{\prime}(2)$ , $f^{\prime}(3)$ .

Determine if the sequence $a_{n}=\frac{2 n^{2}+n-1}{n^{2}}$ converges or diverges, and find the limit if it converges.

Determine the behavior of the curve $x^{2} y - 4 x = 5$ at the point $(5,1)$ : increasing/concave up/down or decreasing/concave up/down?

Find the limit: $\lim _{n \rightarrow \infty} \frac{n^{2}}{\sqrt{n^{3}+4 n}}$ .

Find the derivative of $f(x)=\frac{5}{5x+6}$ . What is $f^{\prime}(x)=?$

Determine if the sequence $a_{n}=\frac{n^{2}}{\sqrt{n^{3}+4 n}}$ converges and find its limit.

Find a relative minimum for the function $h(t)=t-4 \sqrt{1+t}$ using the second derivative test.

Find the derivative of $f(x)=x^{4} \cos x$ . What is $f^{\prime}(x)=?$

Find the critical value of $f(x)=2x+5\cos(x)$ on $0 \leq x \leq 2\pi$ that has absolute and relative extrema. Options: 0, 0.412, 2.730, 6.283.

Find the derivative of the function $f(x)=\left(7 x-5 x^{3}\right)(6+\sqrt{x})$ . What is $f^{\prime}(x)$ ?

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{1}{32+h}-0.03125}{h}$ .

Find the derivative of the function $f(x)=(7x-5x^{3})(6+\sqrt{x})$ . What is $f^{\prime}(x)$ ?

Find $g'(1)$ for $g(x)=2-3x^3$ and use it to write the tangent line at $(1,-1)$ in $y=mx+b$ form.

Find $g(1)$ for $g(x)=2-3x^3$ and use it to find the tangent line at $(1,-1)$ in the form $y=mx+b$ .

Determine if the sequence $a_n = 1 - (0.2)^n$ converges and find its limit.

Deposit \ $1000 at$ 3\%$ simple interest. Find the average rate of change for the first year and six years, to two decimal places.

Find the derivative of the function $f(x)=9 x^{7/5}-5 x^{2}+10^{4}$ .

Find the derivative of the function $f(x)=\cos x-7 \tan x$ , i.e., compute $f^{\prime}(x)$ .

Evaluate the integral: $\int \frac{68 x^{3}+7 x-46}{x^{2}} d x =$

Find the first four terms of the Taylor series for $f(y)=r y \ln \left(\frac{k}{y}\right)$ about $y=K$ .

Find the derivative $f'(x)$ of the function $f(x)=\frac{2-x^{2}}{3+x^{2}}$ and evaluate $f'(3)$ .

Find the derivative $f'(x)$ of the function $f(x)=\frac{\sqrt{x}-2}{\sqrt{x}+2}$ and evaluate $f'(2)$ .

Find the function $f(x)$ and the number $a$ for the limit $\lim _{h \rightarrow 0} \frac{\sqrt{64+h}-8}{h}$ .

Find the limit $\lim _{h \rightarrow 0} \frac{(8+h)^{2}-64}{h}$ and identify the function $f(t)$ and the value $a$ .

Evaluate the integral $\int_{3}^{6} \left(\frac{d}{d t} \sqrt{5+3 t^{4}}\right) d t$ .

Find the derivative $y^{\prime}$ of the function $y=\int_{1}^{\cos (x)}(2 t+10 \sin (t)) d t$ using the Fundamental Theorem of Calculus.

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+7 t+2)(6 t^{-2}+6 t^{-3})$ .

Find the derivative of the function $F(x)=e^{x^{2}}$ . What is $F^{\prime}(x)$ ? $F^{\prime}(x)=\square$

Evaluate the integral $\int_{0}^{4} 2 x e^{x^{2}} d x=\square$ using the Fundamental Theorem of Calculus.

Find the function $f(a)$ and the number $a$ such that $\lim _{h \rightarrow 0} \frac{(8+h)^{2}-64}{h}$ represents its derivative.

Find $f(x)$ where $f'(x)=\frac{\tan(x)}{x}$ and $f(1)=5$ .

If $f(3)=14$ , $f^{\prime}$ is continuous, and $\int_{3}^{7} f^{\prime}(t) dt=20$ , find $f(7)$ .

Find and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{5}{x}$ , then use it to find the derivative $f'(z)$ .

Find and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{5}{x}$ , then use it to find the derivative. Also, find $f^{\prime}(2)$ .

Find the open intervals where the function $f(x)=\cos (x)-\sin (x)$ is increasing on $[0,2 \pi]$ .

Given $f(x)=11 x(\sin (x)+\cos (x))$ , find $f^{\prime}(x)$ and $f^{\prime}\left(\frac{\pi}{4}\right)$ .

Evaluate the integral $\int_{-\pi}^{\pi} f(x) \, dx$ for the piecewise function $f(x) = \{-4x \text{ if } -\pi \leq x \leq 0, \sin(x) \text{ if } 0 < x \leq \pi\}$ .

Find the domain of $y=\frac{1}{x^{2}-3}$ and the largest interval $I$ for the initial condition.

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{-55}^{-5} \frac{d x}{x}=$

Find a function with second derivative $y''=12x-2$ and tangent line $y=-x+5$ at $x=1$ . What is $y=?$

Check if these functions are eigenfunctions of the operator $\frac{d}{d x}$ : a. $e^{-i k x}$ , b. $k x^{2}$ .

Find the value of $\int_{4}^{-4}(f(x)+4) d x$ given $\int_{4}^{8} f(x) d x=7$ and $\int_{-4}^{8} f(x) d x=15$ .

Given $y=5 x^{2}$ , find $\Delta y$ when $x=5$ and $\Delta x=0.1$ . Also, find $d y$ for $d x=0.1$ .

Determine the formula for the integral from 4 to $x^{2}$ of $(t+5)$ with respect to $t$ .

Find the function formula for the integral: $\int_{-2}^{x}(8 t^{2}-7 t) dt =$

Find the derivative $y^{\prime}$ of $y=\int_{1}^{\cos (x)}(-10 t+3 \sin (t)) d t$ using the Fundamental Theorem of Calculus.

Find $d y$ for $y=\tan(3x+3)$ at $x=2$ with $d x=0.1$ and $d x=0.2$ .

Find $F(-9)$ , $F(9)$ , $F^{\prime}(0)$ , and $F^{\prime}(9)$ for $F(x)=\int_{-9}^{x} \frac{d u}{u^{2}+1}$ .