Calculus

Problem 26101

Use the Power Rule for Integration to solve $\int_{4}^{5}\left(\frac{8}{x^{2}}-1\right) d x$ .

See Solution

Problem 26102

Find the equation of the tangent line to $y=5 x^{3}+2 x+3$ at the point $(1,10)$ .

See Solution

Problem 26103

Evaluate the integral from 1 to 3: $\int_{1}^{3} 3 x \, dx = \square$ (Type an integer or a simplified fraction).

See Solution

Problem 26104

Find the intervals where the acceleration of the particle, given by $x(t)=t^{3}-9 t^{2}-21 t$ , is positive.

See Solution

Problem 26105

Find the horizontal asymptote of the function $f(x)=\frac{x^{3}-2 x^{2}+x-3}{x^{2}-17}$ .

See Solution

Problem 26106

Calculate the area of the region enclosed by $y=\sqrt{x}$ , $y=2x-15$ , and $y=0$ .

See Solution

Problem 26107

Find the derivative of the function $y=\frac{2+3 x^{2}}{5 x^{2}-3}$ in simplified form.

See Solution

Problem 26108

Evaluate the integral from 2 to 3: $\int_{2}^{3} 5 x \, dx = \square$ (Type an integer or simplified fraction.)

See Solution

Problem 26109

Find when the particle at $x(t)=t^{3}-48 t^{2}$ moves left for $t \geq 0$ .

See Solution

Problem 26110

Find when the speed of a particle, with position $x(t)=3 t^{3}-18 t^{2}-45 t$ , is increasing for $t \geq 0$ .

See Solution

Problem 26111

Analyze the convergence of the series: $12. \sum_{n=1}^{\infty} e^{-n}$ and $13. \frac{2}{3}-\frac{8}{9}+\frac{32}{27}-\frac{128}{81}+\ldots$ . If convergent, find the sum.

See Solution

Problem 26112

Find the change in $F(x)=9x+130$ from $x=14$ to $x=19$ and verify it with the area under $F'(x)$ from $x=14$ to $x=19$ .

See Solution

Problem 26113

Calculate the integral from 2 to 6 of the constant 8: $\int_{2}^{6} 8 d x = \square$ .

See Solution

Problem 26114

Evaluate the integral from 2 to 5 of $(2x + 6)$ . What is the result? $\int_{2}^{5}(2 x+6) d x=\square$

See Solution

Problem 26115

Evaluate the integral from 0 to 5: $\int_{0}^{5} e^{2 x} d x = \square$ . Round to the nearest thousandth.

See Solution

Problem 26116

Find when the acceleration of a particle, with velocity $v(t)=6 t^{2}-24 t$ , is positive for $t \geq 0$ .

See Solution

Problem 26117

Evaluate the integral from 0 to 3: $\int_{0}^{3} e^{3 x} d x = \square$ (round to the nearest thousandth).

See Solution

Problem 26118

Evaluate the integral: $\int_{3}^{6} \frac{5}{x} d x=\square$ (Provide the exact answer.)

See Solution

Problem 26119

Calculate the integral from 2 to 3 of (2x + 1): $\int_{2}^{3}(2 x+1) d x=\square$

See Solution

Problem 26120

Evaluate the integral from 6 to 1: $\int_{6}^{1}(2 x+3) d x=\square($ Simplify your answer. $)$

See Solution

Problem 26121

Find when the speed of a particle, given by $x(t)=4 t^{3}-48 t^{2}+84 t$ , is decreasing for $t \geq 0$ .

See Solution

Problem 26122

Find the integral: $\int \frac{\sec ^{2} x}{\tan ^{3} x-3 \tan ^{2} x} d x$

See Solution

Problem 26123

Evaluate the integral from 3 to 4: $\int_{3}^{4}\left(4 x^{3}+8\right) d x=\square$ (Type an integer or a simplified fraction.)

See Solution

Problem 26124

Evaluate the integral from 1 to 4 of $(6 x^{-2}-3)$ . What is the exact simplified answer?

See Solution

Problem 26125

Calculate the integral $\int_{0}^{1} 4 \sqrt[4]{x} \, dx = \square$ (provide the exact simplified answer).

See Solution

Problem 26126

Find intervals where the particle moves right given its position $x(t)=3 t^{3}-36 t^{2}-81 t$ for $t \geq 0$ .

See Solution

Problem 26127

Find the average value of $f(x)=-6 x^{2}-5$ on $[0,4]$ . The average value is $\square$ .

See Solution

Problem 26128

Find the average value of $f(x)=\sqrt[4]{x}$ over $[1,16]$ and graph it with the function. Average value: $\square$ .

See Solution

Problem 26129

Determine the radius and interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \cdot(x-1)^{n}$ .

See Solution

Problem 26130

Find the intervals where the speed of the particle, described by $x(t)=4 t^{3}-24 t^{2}+48 t$ , is increasing.

See Solution

Problem 26131

Find the area in the first quadrant between $y=9$ and $y=9 \sin x$ from $x=0$ to $x=\frac{\pi}{2}$ .

See Solution

Problem 26132

Find the average value of $f(x)=600-90x$ over $[0,5]$ . The average value is $\square$ .

See Solution

Problem 26133

Find the cost increase when producing 300 to 450 bikes using the marginal cost function $C^{\prime}(x)=500-\frac{x}{3}$ .

See Solution

Problem 26134

Find the change in $F(x)=4x+130$ from $x=8$ to $x=18$ and verify it with the area under $F'(x)$ from $x=8$ to $x=18$ .

See Solution

Problem 26135

Find the average value of $f(x)=-4 x^{2}-3$ on $[0,3]$ and graph it with the function in the same window.

See Solution

Problem 26136

Solve the ODE $2 y^{\prime \prime}-3 y^{\prime}+y=\cos (t / 2)$ with $y(0)=1$ , $y^{\prime}(2)=5$ . Which method can't be used?

See Solution

Problem 26137

Use the shooting method for the ODE $(2x-3)y'''-(6x-7)y''+4xy'-4y=0$ with conditions $y(0)=0$ , $y'(0)=0$ , $y(1)+y'(1)=1$ . Which statement is correct?

See Solution

Problem 26138

Determine if the particle with velocity $v(t)=-3 t^{2}+4 t+15$ is speeding up or slowing down at $t=2$ .

See Solution

Problem 26139

Check if Rolle's theorem applies to $f(x)=4-x^{2/3}$ on $[-1,1]$ and find the guaranteed point(s).

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Problem 26140

Calculate $\int_{3}^{5}(7x+x^{2})dx$ using given integrals: $\int_{3}^{5} x^{2} dx=\frac{98}{3}$ , $\int_{3}^{5} x dx=8$ .

See Solution

Problem 26141

Check if Rolle's Theorem applies to $h(x)=e^{x^{2}}$ on $[-a, a]$ for $a>0$ and find guaranteed point(s).

See Solution

Problem 26142

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty}(-1)^{n} n^{4}(x+4)^{n}$ .

See Solution

Problem 26143

Calculate $\int_{1}^{7} 6 x^{2} d x$ using $\int_{1}^{6} x^{2} d x=\frac{215}{3}$ and $\int_{6}^{7} x^{2} d x=\frac{127}{3}$ .

See Solution

Problem 26144

Calculate the integral $\int_{5}^{2} 8 x^{2} d x$ using given values: $\int_{1}^{2} x d x=1.5$ , $\int_{1}^{2} x^{2} d x=\frac{7}{3}$ , $\int_{2}^{5} x^{2} d x=39$ .

See Solution

Problem 26145

Is it true that if $\int^{b} f(x) dx=0$ , then $f(x)=0$ for all $x$ in $[a, b]$ ? Explain or provide a counterexample.

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Problem 26146

Find the point of maximum growth rate for the function $f(x)=\frac{20}{1+9 e^{-3 x}}$ . Round to the nearest hundredth.

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Problem 26147

Check if the Mean Value Theorem applies to $f(x)=\sin^{-1} x$ on $[-\frac{\sqrt{3}}{2}, 0]$ and find the guaranteed point(s).

See Solution

Problem 26148

An object dropped from $256 \mathrm{ft}$ has height $s(t)=256-16 t^{2}$ . Find its velocity and acceleration when it hits the ground.

See Solution

Problem 26149

Check if the Mean Value Theorem applies to $f(x)=e^{x}$ on $[0, \ln 15]$ and find the guaranteed point(s).

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Problem 26150

Check if the Mean Value Theorem applies to $f(x)=-6+x^{2}$ on $[-1,2]$ and find the guaranteed point(s).

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Problem 26151

Estimate the area under $N(t)$ from $t=0$ to $t=60$ using left and right sums with 3 equal subintervals. Calculate error bounds.

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Problem 26152

A factory discharges pollution at $r(t)=\frac{t}{t^{2}+1}$ . Find total pollution in 9 years. Round to two decimals.

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Problem 26153

Check if the Mean Value Theorem applies to $f(x)=\ln 3x$ on $[1, e]$ and find the guaranteed point(s).

See Solution

Problem 26154

Solve the equation $y^{4} y' = 8 x^{7}$ with $y(0) = 1$ and verify your solution.

See Solution

Problem 26155

Find the interval where the growth rate of $f(x)=\frac{24}{1+3 e^{-1.3 x}}$ is decreasing.

See Solution

Problem 26156

Solve the equation $y^{4} y' = 8 x^{7}$ with the initial condition $y(0) = 1$ and verify your solution.

See Solution

Problem 26157

Identify which functions among $g(x)=5 x^{10}$ , $h(x)=x^{10}+5$ , or $p(x)=x^{10}-\ln 5$ share the same derivative as $f(x)=x^{10}$ .

See Solution

Problem 26158

Find a power series that converges for $x$ in the interval $[2,6)$ .

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Problem 26159

Find the derivative of the function $y = x(5 - 2x)(3 - 2x)$ . What is $y'$ ?

See Solution

Problem 26160

Find power series for $f(x)=\frac{x}{9+x^{2}}$ and $g(x)=\arctan \left(\frac{x}{3}\right)$ , then determine convergence intervals.

See Solution

Problem 26161

Find the interval where the growth rate of $f(x)=\frac{24}{1+3 e^{-1.3 x}}$ is decreasing. Options:
1. $\left(\frac{\ln 1.3}{3}, \infty\right)$
2. $\left(-\infty, \frac{\ln 3}{1.3}\right)$
3. $\left(\frac{\ln 3}{1.3}, \infty\right)$
4. $\left(-\infty, \frac{\ln 1.3}{3}\right)$

See Solution

Problem 26162

Evaluate the integral: $\int \frac{x+1}{x \sqrt{16+x^{2}}} d x$ with the constant of integration $C$ .

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Problem 26163

Find all values of $c$ in $(0, \pi)$ for the function $f(x) = 2\sin x + \sin 2x$ that satisfy MVT.

See Solution

Problem 26164

A box is made from a 3M x 5M cardboard sheet. Find the max volume and critical points for $y = 14x^3 - 16x^2 + 15x$ .

See Solution

Problem 26165

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x) = \frac{2}{x-1}$ .

See Solution

Problem 26166

Use Newton's Law of Cooling: $T(t)=T_{A}+(T_{0}-T_{A}) e^{-k t}$ .
Mr. Mearig's coffee cools from $200^{\circ} F$ to $180^{\circ} F$ in 7 minutes at $72^{\circ} F$ .
(a) Find $k$ (round to the nearest thousandth). (b) How long to reach $140^{\circ} F$ using $k$ from (a)? Round to the nearest minute.
BONUS: Discuss how a smaller mug affects $k$ for faster cooling.

See Solution

Problem 26167

Find the total basal metabolism, $\int_{0}^{24} R(t) dt$ , for $R(t)=60-0.18 \cos \left(\frac{\pi t}{12}\right)$ over 24 hours.

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Problem 26168

Solve the equation $2 y^{\prime \prime} + y^{\prime} - y = -1$ with $y(0) = 1$ and $y^{\prime}(0) = 2$ .

See Solution

Problem 26169

Calculate the sums: $(-\frac{2}{3}) + (-\frac{2}{3})^2 + (-\frac{2}{3})^3 + \ldots$ and $\sum_{j=1}^{\infty}(5)^{j}$ .

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Problem 26170

Compute the sums:
1. $2\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right)^{2} + 2\left(\frac{1}{2}\right)^{3} + \ldots$
2. $\sum_{j=1}^{\infty} 4\left(-\frac{3}{4}\right)^{j}$
Provide exact values or state "No sum".

See Solution

Problem 26171

Calculate the sums:
1. $5 + 5\left(-\frac{1}{2}\right) + 5\left(-\frac{1}{2}\right)^{2} + \ldots = \square$
2. $\sum_{k=1}^{\infty} 3\left(\frac{4}{7}\right)^{k} = \square$

See Solution

Problem 26172

Find the power series for $\ln(1+x)$ , its interval of convergence, and the sum of the series $\frac{1}{1 \cdot 2}-\frac{1}{2 \cdot 2^{2}}+\cdots$ .

See Solution

Problem 26173

Find the derivative of $y=\cos^{-1}\left(\frac{5}{x}\right)$ with respect to $x$ .

See Solution

Problem 26174

Find the Maclaurin series for $\sin x$ , deduce for $\cos x$ , find terms for $\tan x$ , and evaluate $\lim_{x \to 0} \frac{\tan x - x}{x^3}$ .

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Problem 26175

A satellite of mass $m=1285 \mathrm{~kg}$ falls from $R=2.2$ Earth radii. Find its speed just before hitting the Earth.

See Solution

Problem 26176

Find power series for $f(x)=\frac{x}{9+x^{2}}$ and $g(x)=\arctan \left(\frac{x}{3}\right)$ , including intervals of convergence.

See Solution

Problem 26177

Find the number of ODEs from the PDE $\alpha^{2} u_{x x x x}=u_{t t}$ using separation of variables. A. Six. B. Four. C. Two. D. None.

See Solution

Problem 26178

Find the tangent to the cycloid $x=r(\theta-\sin \theta), y=r(1-\cos \theta)$ at $\theta=\pi/3$ . When is the tangent horizontal or vertical?

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Problem 26179

Solve Euler's equation $a x^{2} y^{\prime \prime}+(b+a) x y^{\prime}+c y=0$ with $b^{2}-4 a c=-1$ . Find $y(x)$ .

See Solution

Problem 26180

Find the Laplace Transform of $f(t)=e^{-at} u_{b}(t)$ where $a, b$ are constants. Choose the correct option.

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Problem 26181

Solve Euler's equation $a x^{2} y^{\prime \prime}+(b+a) x y^{\prime}+c y=0$ with $y(x)=x^{k}$ , $x=e^{t}$ , $b^{2}-4 a c=-1$ .

See Solution

Problem 26182

Solve the equation $a x^{2} y^{\prime \prime}+(b+a) x y^{\prime}+c y=0$ with $y(x)=x^{k}$ and $x=e^{t}$ , given $b^{2}-4 a c=1$ .

See Solution

Problem 26183

Solve $2 y^{\prime \prime}-y^{\prime}-y=1$ with $y(0)=1$ and $y^{\prime}(0)=-2$ . Find $y^{\prime \prime \prime}(0)$ .

See Solution

Problem 26184

Solve the equation $2 y^{\prime \prime}-y^{\prime}-y=1$ with $y(0)=1$ and $y^{\prime}(0)=-2$ . Find $y^{\prime \prime \prime}(0)$ .

See Solution

Problem 26185

Classify the ODE: $\frac{d y}{d x}=\frac{y(1-6 x)}{x(2-7 y)}$ . Options: first order, non-linear, homogeneous; first order, linear, non-homogeneous; first order, non-linear, non-homogeneous; first order, linear, homogeneous.

See Solution

Problem 26186

Evaluate the integral $\int_{2}^{3}\left(\frac{d}{d t} \sqrt{4+5 t^{4}}\right) d t$ using the Fundamental Theorem of Calculus.

See Solution

Problem 26187

Find the Laplace Transform of $f(t)=t$ for $0 \leq t<1$ and $f(t)=0$ for $t \geq 1$ . What is $L\{f(t)\}$ ?

See Solution

Problem 26188

Find the correct coefficients $a_n$ and $b_n$ for the function $u(x, t)$ given the initial conditions. Options are provided.

See Solution

Problem 26189

Find the Laplace Transform of $f(t)=e^{-at} u_b(t)$ where $a, b$ are constants.

See Solution

Problem 26190

Find the maximum volume of a box made from a $3 \mathrm{~m} \times 5 \mathrm{~m}$ cardboard sheet by cutting squares from corners.

See Solution

Problem 26191

Find the Taylor series for $f(x)=x e^{2 x}$ at $a=0$ and $g(x)=\frac{1}{x}$ at $a=-3$ , and determine their radii of convergence.

See Solution

Problem 26192

Evaluate the integral $\int_{0}^{1} \frac{7}{t^{2}+1} d t$ using the Fundamental Theorem of Calculus.

See Solution

Problem 26193

Classify the ODE: $\frac{d y}{d x}=\frac{y(5-6 x)}{x(5-7 y)}$ . Choose: first order, non-linear, homogeneous/non-homogeneous/linear.

See Solution

Problem 26194

Determine where the function $f(x)=4 x^{2} \ln x^{2}+4$ is increasing or decreasing.

See Solution

Problem 26195

Given the PDE $\alpha^{2} u_{x x x x}=u_{t t}$ , how many ODEs arise from separation of variables $u(x, t)=X(x) T(t)$ ? A. Six. B. Two. C. Four. D. None of the above.

See Solution

Problem 26196

Show that $\int_{0}^{\pi / 2} \sum_{n=1}^{\infty} \frac{\cos n x}{n^{2}} d x=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1)^{3}}$ . Prove $\sum_{n=1}^{\infty} f_{n}(x)$ lacks property $\mathscr{C}$ on $[0,1]$ .

See Solution

Problem 26197

Classify the differential equation: $\frac{d^{2} x}{d t^{2}}+4 \frac{d x}{d t}+9 x=4 \cos 4 t$ . Options: 2nd order, linear, non-homogeneous; 2nd order, linear, homogeneous; 2nd order, non-linear, homogeneous; 2nd order, non-linear, non-homogeneous.

See Solution

Problem 26198

Calculate the infinite sum $\sum_{i=1}^{\infty} \frac{4}{7}\left(\frac{7}{6}\right)^{i-1}$ .

See Solution

Problem 26199

Classify the ODE: $\frac{d y}{d x}=\frac{y(1-6 x)}{x(2-7 y)}$ . Choose: 1st order, non-linear, homogeneous or not?

See Solution

Problem 26200

Find the Laplace Transform of $f(t)=\begin{cases}t, & 0 \leq t<1 \\ 0, & t \geq 1\end{cases}$ .

See Solution

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Calculus

Problem 26101

Use the Power Rule for Integration to solve ∫45(8x2−1)dx\int_{4}^{5}\left(\frac{8}{x^{2}}-1\right) d x∫45​(x28​−1)dx.

Problem 26102

Find the equation of the tangent line to y=5x3+2x+3y=5 x^{3}+2 x+3y=5x3+2x+3 at the point (1,10)(1,10)(1,10).

Problem 26103

Evaluate the integral from 1 to 3: ∫133x dx=□\int_{1}^{3} 3 x \, dx = \square∫13​3xdx=□ (Type an integer or a simplified fraction).

Problem 26104

Find the intervals where the acceleration of the particle, given by x(t)=t3−9t2−21tx(t)=t^{3}-9 t^{2}-21 tx(t)=t3−9t2−21t, is positive.

Problem 26105

Find the horizontal asymptote of the function f(x)=x3−2x2+x−3x2−17f(x)=\frac{x^{3}-2 x^{2}+x-3}{x^{2}-17}f(x)=x2−17x3−2x2+x−3​.

Problem 26106

Calculate the area of the region enclosed by y=xy=\sqrt{x}y=x​, y=2x−15y=2x-15y=2x−15, and y=0y=0y=0.

Problem 26107

Find the derivative of the function y=2+3x25x2−3y=\frac{2+3 x^{2}}{5 x^{2}-3}y=5x2−32+3x2​ in simplified form.

Problem 26108

Evaluate the integral from 2 to 3: ∫235x dx=□\int_{2}^{3} 5 x \, dx = \square∫23​5xdx=□ (Type an integer or simplified fraction.)

Problem 26109

Find when the particle at x(t)=t3−48t2x(t)=t^{3}-48 t^{2}x(t)=t3−48t2 moves left for t≥0t \geq 0t≥0.

Problem 26110

Find when the speed of a particle, with position x(t)=3t3−18t2−45tx(t)=3 t^{3}-18 t^{2}-45 tx(t)=3t3−18t2−45t, is increasing for t≥0t \geq 0t≥0.

Problem 26111

Analyze the convergence of the series: 12.∑n=1∞e−n12. \sum_{n=1}^{\infty} e^{-n}12.∑n=1∞​e−n and 13.23−89+3227−12881+…13. \frac{2}{3}-\frac{8}{9}+\frac{32}{27}-\frac{128}{81}+\ldots13.32​−98​+2732​−81128​+…. If convergent, find the sum.

Problem 26112

Find the change in F(x)=9x+130F(x)=9x+130F(x)=9x+130 from x=14x=14x=14 to x=19x=19x=19 and verify it with the area under F′(x)F'(x)F′(x) from x=14x=14x=14 to x=19x=19x=19.

Problem 26113

Calculate the integral from 2 to 6 of the constant 8: ∫268dx=□\int_{2}^{6} 8 d x = \square∫26​8dx=□.

Problem 26114

Evaluate the integral from 2 to 5 of (2x+6)(2x + 6)(2x+6). What is the result? ∫25(2x+6)dx=□\int_{2}^{5}(2 x+6) d x=\square∫25​(2x+6)dx=□

Problem 26115

Evaluate the integral from 0 to 5: ∫05e2xdx=□\int_{0}^{5} e^{2 x} d x = \square∫05​e2xdx=□. Round to the nearest thousandth.

Problem 26116

Find when the acceleration of a particle, with velocity v(t)=6t2−24tv(t)=6 t^{2}-24 tv(t)=6t2−24t, is positive for t≥0t \geq 0t≥0.

Problem 26117

Evaluate the integral from 0 to 3: ∫03e3xdx=□\int_{0}^{3} e^{3 x} d x = \square∫03​e3xdx=□ (round to the nearest thousandth).

Problem 26118

Evaluate the integral: ∫365xdx=□\int_{3}^{6} \frac{5}{x} d x=\square∫36​x5​dx=□ (Provide the exact answer.)

Problem 26119

Calculate the integral from 2 to 3 of (2x + 1): ∫23(2x+1)dx=□\int_{2}^{3}(2 x+1) d x=\square∫23​(2x+1)dx=□

Problem 26120

Evaluate the integral from 6 to 1: ∫61(2x+3)dx=□(\int_{6}^{1}(2 x+3) d x=\square(∫61​(2x+3)dx=□( Simplify your answer. )))

Problem 26121

Find when the speed of a particle, given by x(t)=4t3−48t2+84tx(t)=4 t^{3}-48 t^{2}+84 tx(t)=4t3−48t2+84t, is decreasing for t≥0t \geq 0t≥0.

Problem 26122

Find the integral: ∫sec⁡2xtan⁡3x−3tan⁡2xdx\int \frac{\sec ^{2} x}{\tan ^{3} x-3 \tan ^{2} x} d x∫tan3x−3tan2xsec2x​dx

Problem 26123

Evaluate the integral from 3 to 4: ∫34(4x3+8)dx=□\int_{3}^{4}\left(4 x^{3}+8\right) d x=\square∫34​(4x3+8)dx=□ (Type an integer or a simplified fraction.)

Problem 26124

Evaluate the integral from 1 to 4 of (6x−2−3)(6 x^{-2}-3)(6x−2−3). What is the exact simplified answer?

Problem 26125

Calculate the integral ∫014x4 dx=□\int_{0}^{1} 4 \sqrt[4]{x} \, dx = \square∫01​44x​dx=□ (provide the exact simplified answer).

Problem 26126

Find intervals where the particle moves right given its position x(t)=3t3−36t2−81tx(t)=3 t^{3}-36 t^{2}-81 tx(t)=3t3−36t2−81t for t≥0t \geq 0t≥0.

Problem 26127

Find the average value of f(x)=−6x2−5f(x)=-6 x^{2}-5f(x)=−6x2−5 on [0,4][0,4][0,4]. The average value is □\square□.

Problem 26128

Find the average value of f(x)=x4f(x)=\sqrt[4]{x}f(x)=4x​ over [1,16][1,16][1,16] and graph it with the function. Average value: □\square□.

Problem 26129

Determine the radius and interval of convergence for the series ∑n=1∞(−1)n+1n⋅(x−1)n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \cdot(x-1)^{n}∑n=1∞​n(−1)n+1​⋅(x−1)n.

Problem 26130

Find the intervals where the speed of the particle, described by x(t)=4t3−24t2+48tx(t)=4 t^{3}-24 t^{2}+48 tx(t)=4t3−24t2+48t, is increasing.

Problem 26131

Find the area in the first quadrant between y=9y=9y=9 and y=9sin⁡xy=9 \sin xy=9sinx from x=0x=0x=0 to x=π2x=\frac{\pi}{2}x=2π​.

Problem 26132

Find the average value of f(x)=600−90xf(x)=600-90xf(x)=600−90x over [0,5][0,5][0,5]. The average value is □\square□.

Problem 26133

Find the cost increase when producing 300 to 450 bikes using the marginal cost function C′(x)=500−x3C^{\prime}(x)=500-\frac{x}{3}C′(x)=500−3x​.

Problem 26134

Find the change in F(x)=4x+130F(x)=4x+130F(x)=4x+130 from x=8x=8x=8 to x=18x=18x=18 and verify it with the area under F′(x)F'(x)F′(x) from x=8x=8x=8 to x=18x=18x=18.

Problem 26135

Find the average value of f(x)=−4x2−3f(x)=-4 x^{2}-3f(x)=−4x2−3 on [0,3][0,3][0,3] and graph it with the function in the same window.

Problem 26136

Solve the ODE 2y′′−3y′+y=cos⁡(t/2)2 y^{\prime \prime}-3 y^{\prime}+y=\cos (t / 2)2y′′−3y′+y=cos(t/2) with y(0)=1y(0)=1y(0)=1, y′(2)=5y^{\prime}(2)=5y′(2)=5. Which method can't be used?

Problem 26137

Use the shooting method for the ODE (2x−3)y′′′−(6x−7)y′′+4xy′−4y=0(2x-3)y'''-(6x-7)y''+4xy'-4y=0(2x−3)y′′′−(6x−7)y′′+4xy′−4y=0 with conditions y(0)=0y(0)=0y(0)=0, y′(0)=0y'(0)=0y′(0)=0, y(1)+y′(1)=1y(1)+y'(1)=1y(1)+y′(1)=1. Which statement is correct?

Problem 26138

Determine if the particle with velocity v(t)=−3t2+4t+15v(t)=-3 t^{2}+4 t+15v(t)=−3t2+4t+15 is speeding up or slowing down at t=2t=2t=2.

Problem 26139

Check if Rolle's theorem applies to f(x)=4−x2/3f(x)=4-x^{2/3}f(x)=4−x2/3 on [−1,1][-1,1][−1,1] and find the guaranteed point(s).

Problem 26140

Use the Power Rule for Integration to solve $\int_{4}^{5}\left(\frac{8}{x^{2}}-1\right) d x$ .

Find the equation of the tangent line to $y=5 x^{3}+2 x+3$ at the point $(1,10)$ .

Evaluate the integral from 1 to 3: $\int_{1}^{3} 3 x \, dx = \square$ (Type an integer or a simplified fraction).

Find the intervals where the acceleration of the particle, given by $x(t)=t^{3}-9 t^{2}-21 t$ , is positive.

Find the horizontal asymptote of the function $f(x)=\frac{x^{3}-2 x^{2}+x-3}{x^{2}-17}$ .

Calculate the area of the region enclosed by $y=\sqrt{x}$ , $y=2x-15$ , and $y=0$ .

Find the derivative of the function $y=\frac{2+3 x^{2}}{5 x^{2}-3}$ in simplified form.

Evaluate the integral from 2 to 3: $\int_{2}^{3} 5 x \, dx = \square$ (Type an integer or simplified fraction.)

Find when the particle at $x(t)=t^{3}-48 t^{2}$ moves left for $t \geq 0$ .

Find when the speed of a particle, with position $x(t)=3 t^{3}-18 t^{2}-45 t$ , is increasing for $t \geq 0$ .

Analyze the convergence of the series: $12. \sum_{n=1}^{\infty} e^{-n}$ and $13. \frac{2}{3}-\frac{8}{9}+\frac{32}{27}-\frac{128}{81}+\ldots$ . If convergent, find the sum.

Find the change in $F(x)=9x+130$ from $x=14$ to $x=19$ and verify it with the area under $F'(x)$ from $x=14$ to $x=19$ .

Calculate the integral from 2 to 6 of the constant 8: $\int_{2}^{6} 8 d x = \square$ .

Evaluate the integral from 2 to 5 of $(2x + 6)$ . What is the result? $\int_{2}^{5}(2 x+6) d x=\square$

Evaluate the integral from 0 to 5: $\int_{0}^{5} e^{2 x} d x = \square$ . Round to the nearest thousandth.

Find when the acceleration of a particle, with velocity $v(t)=6 t^{2}-24 t$ , is positive for $t \geq 0$ .

Evaluate the integral from 0 to 3: $\int_{0}^{3} e^{3 x} d x = \square$ (round to the nearest thousandth).

Evaluate the integral: $\int_{3}^{6} \frac{5}{x} d x=\square$ (Provide the exact answer.)

Calculate the integral from 2 to 3 of (2x + 1): $\int_{2}^{3}(2 x+1) d x=\square$

Evaluate the integral from 6 to 1: $\int_{6}^{1}(2 x+3) d x=\square($ Simplify your answer. $)$

Find when the speed of a particle, given by $x(t)=4 t^{3}-48 t^{2}+84 t$ , is decreasing for $t \geq 0$ .

Find the integral: $\int \frac{\sec ^{2} x}{\tan ^{3} x-3 \tan ^{2} x} d x$

Evaluate the integral from 3 to 4: $\int_{3}^{4}\left(4 x^{3}+8\right) d x=\square$ (Type an integer or a simplified fraction.)

Evaluate the integral from 1 to 4 of $(6 x^{-2}-3)$ . What is the exact simplified answer?

Calculate the integral $\int_{0}^{1} 4 \sqrt[4]{x} \, dx = \square$ (provide the exact simplified answer).

Find intervals where the particle moves right given its position $x(t)=3 t^{3}-36 t^{2}-81 t$ for $t \geq 0$ .

Find the average value of $f(x)=-6 x^{2}-5$ on $[0,4]$ . The average value is $\square$ .

Find the average value of $f(x)=\sqrt[4]{x}$ over $[1,16]$ and graph it with the function. Average value: $\square$ .

Determine the radius and interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \cdot(x-1)^{n}$ .

Find the intervals where the speed of the particle, described by $x(t)=4 t^{3}-24 t^{2}+48 t$ , is increasing.

Find the area in the first quadrant between $y=9$ and $y=9 \sin x$ from $x=0$ to $x=\frac{\pi}{2}$ .

Find the average value of $f(x)=600-90x$ over $[0,5]$ . The average value is $\square$ .

Find the cost increase when producing 300 to 450 bikes using the marginal cost function $C^{\prime}(x)=500-\frac{x}{3}$ .

Find the change in $F(x)=4x+130$ from $x=8$ to $x=18$ and verify it with the area under $F'(x)$ from $x=8$ to $x=18$ .

Find the average value of $f(x)=-4 x^{2}-3$ on $[0,3]$ and graph it with the function in the same window.

Solve the ODE $2 y^{\prime \prime}-3 y^{\prime}+y=\cos (t / 2)$ with $y(0)=1$ , $y^{\prime}(2)=5$ . Which method can't be used?

Use the shooting method for the ODE $(2x-3)y'''-(6x-7)y''+4xy'-4y=0$ with conditions $y(0)=0$ , $y'(0)=0$ , $y(1)+y'(1)=1$ . Which statement is correct?

Determine if the particle with velocity $v(t)=-3 t^{2}+4 t+15$ is speeding up or slowing down at $t=2$ .

Check if Rolle's theorem applies to $f(x)=4-x^{2/3}$ on $[-1,1]$ and find the guaranteed point(s).

Calculate $\int_{3}^{5}(7x+x^{2})dx$ using given integrals: $\int_{3}^{5} x^{2} dx=\frac{98}{3}$ , $\int_{3}^{5} x dx=8$ .

Check if Rolle's Theorem applies to $h(x)=e^{x^{2}}$ on $[-a, a]$ for $a>0$ and find guaranteed point(s).

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty}(-1)^{n} n^{4}(x+4)^{n}$ .

Calculate $\int_{1}^{7} 6 x^{2} d x$ using $\int_{1}^{6} x^{2} d x=\frac{215}{3}$ and $\int_{6}^{7} x^{2} d x=\frac{127}{3}$ .

Calculate the integral $\int_{5}^{2} 8 x^{2} d x$ using given values: $\int_{1}^{2} x d x=1.5$ , $\int_{1}^{2} x^{2} d x=\frac{7}{3}$ , $\int_{2}^{5} x^{2} d x=39$ .

Is it true that if $\int^{b} f(x) dx=0$ , then $f(x)=0$ for all $x$ in $[a, b]$ ? Explain or provide a counterexample.

Find the point of maximum growth rate for the function $f(x)=\frac{20}{1+9 e^{-3 x}}$ . Round to the nearest hundredth.

Check if the Mean Value Theorem applies to $f(x)=\sin^{-1} x$ on $[-\frac{\sqrt{3}}{2}, 0]$ and find the guaranteed point(s).

An object dropped from $256 \mathrm{ft}$ has height $s(t)=256-16 t^{2}$ . Find its velocity and acceleration when it hits the ground.

Check if the Mean Value Theorem applies to $f(x)=e^{x}$ on $[0, \ln 15]$ and find the guaranteed point(s).

Check if the Mean Value Theorem applies to $f(x)=-6+x^{2}$ on $[-1,2]$ and find the guaranteed point(s).

Estimate the area under $N(t)$ from $t=0$ to $t=60$ using left and right sums with 3 equal subintervals. Calculate error bounds.

A factory discharges pollution at $r(t)=\frac{t}{t^{2}+1}$ . Find total pollution in 9 years. Round to two decimals.

Check if the Mean Value Theorem applies to $f(x)=\ln 3x$ on $[1, e]$ and find the guaranteed point(s).

Solve the equation $y^{4} y' = 8 x^{7}$ with $y(0) = 1$ and verify your solution.

Find the interval where the growth rate of $f(x)=\frac{24}{1+3 e^{-1.3 x}}$ is decreasing.

Solve the equation $y^{4} y' = 8 x^{7}$ with the initial condition $y(0) = 1$ and verify your solution.

Identify which functions among $g(x)=5 x^{10}$ , $h(x)=x^{10}+5$ , or $p(x)=x^{10}-\ln 5$ share the same derivative as $f(x)=x^{10}$ .

Find a power series that converges for $x$ in the interval $[2,6)$ .

Find the derivative of the function $y = x(5 - 2x)(3 - 2x)$ . What is $y'$ ?

Find power series for $f(x)=\frac{x}{9+x^{2}}$ and $g(x)=\arctan \left(\frac{x}{3}\right)$ , then determine convergence intervals.

Evaluate the integral: $\int \frac{x+1}{x \sqrt{16+x^{2}}} d x$ with the constant of integration $C$ .

Find all values of $c$ in $(0, \pi)$ for the function $f(x) = 2\sin x + \sin 2x$ that satisfy MVT.

A box is made from a 3M x 5M cardboard sheet. Find the max volume and critical points for $y = 14x^3 - 16x^2 + 15x$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x) = \frac{2}{x-1}$ .

Find the total basal metabolism, $\int_{0}^{24} R(t) dt$ , for $R(t)=60-0.18 \cos \left(\frac{\pi t}{12}\right)$ over 24 hours.

Solve the equation $2 y^{\prime \prime} + y^{\prime} - y = -1$ with $y(0) = 1$ and $y^{\prime}(0) = 2$ .

Calculate the sums: $(-\frac{2}{3}) + (-\frac{2}{3})^2 + (-\frac{2}{3})^3 + \ldots$ and $\sum_{j=1}^{\infty}(5)^{j}$ .

Calculate the sums:
1. $5 + 5\left(-\frac{1}{2}\right) + 5\left(-\frac{1}{2}\right)^{2} + \ldots = \square$
2. $\sum_{k=1}^{\infty} 3\left(\frac{4}{7}\right)^{k} = \square$

Find the power series for $\ln(1+x)$ , its interval of convergence, and the sum of the series $\frac{1}{1 \cdot 2}-\frac{1}{2 \cdot 2^{2}}+\cdots$ .

Find the derivative of $y=\cos^{-1}\left(\frac{5}{x}\right)$ with respect to $x$ .

Find the Maclaurin series for $\sin x$ , deduce for $\cos x$ , find terms for $\tan x$ , and evaluate $\lim_{x \to 0} \frac{\tan x - x}{x^3}$ .