Calculus

Problem 13801

Find the dimensions of a rectangular tank with a square base and a volume of $37,044 \mathrm{ft}^{3}$ that minimizes surface area. The height is $\square \mathrm{ft}$ and the base side length is $\square \mathrm{ft}$ .

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Problem 13802

Analyze the Leonard-Jones 6-3 potential: $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ . What happens as $r \rightarrow 0$ ?

See Solution

Problem 13803

Analyze the function $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ for $r>0$ .
(a) Determine the limits as $r \rightarrow 0$ and $r \rightarrow \infty$ .
(b) Discuss the existence of a minimum for $V(r)$ .

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Problem 13804

Given the function $f(x)=x^{2}-1$ , find the derivative at $x=2$ , $f(2)$ , tangent line equation, and graph both.

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Problem 13805

Use the function $f(x)=x^{2}-1$ to find: (a) the derivative at $x=2$ , (b) $f(2)$ and tangent line, (c) graph both.

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Problem 13806

Analyze the function $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ for $r>0$ .
(a) What happens to $V(r)$ as $r \rightarrow 0$ and $r \rightarrow \infty$ ? (b) Explain if a minimum exists and find that minimum value of $r$ . $r=\square$

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Problem 13807

What is the limit of $V(r)=e^{-r}-A e^{-a r}$ as $r \rightarrow 0$ for $a=\frac{1}{2}$ and $A=1$ ?

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Problem 13808

Analyze the function $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ for $r>0$ . What happens as $r \to 0$ and $r \to \infty$ ?

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Problem 13809

Find conditions for a polynomial to have an absolute minimum. Analyze a graph to find its lowest degree and concave down interval.

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Problem 13810

Fish interactions are modeled by the Morse potential $V(r)=e^{-r}-A e^{-a r}$ . With $a=\frac{1}{2}$ and $A=1$ , analyze $V(r)$ as $r \rightarrow 0$ and $\infty$ , then find $r$ that minimizes $V(r)$ . $r=\square$

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Problem 13811

Given the function $f(x)=x^{2}+1$ , find the derivative at $x=2$ , $f(2)$ , the tangent line, and graph them.

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Problem 13812

Analyze the function $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ for behavior as $r \to \infty$ , find minimizing $r$ , and consider $A<0$ .

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Problem 13813

Analyze the function $V(r)=e^{-r}-A e^{-a r}$ for $a=\frac{1}{2}$ , $A=1$ : behavior as $r \to 0$ and $r \to \infty$ . Find $r$ that minimizes $V(r)$ .

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Problem 13814

Analyze the function $V(r)=e^{-r}-A e^{-a r}$ for $a=\frac{1}{2}$ and $A=1$ . What happens as $r \rightarrow 0$ and $r \rightarrow \infty$ ?

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Problem 13815

Analyze the function $V(r)=e^{-r}-A e^{-a r}$ for $a=\frac{1}{2}$ , $A=1$ and find its minimum value for $r>0$ .

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Problem 13816

Given the function $V(r)=e^{-r}-Ae^{-\frac{1}{2}r}$ for $r>0$ , analyze its behavior as $r \rightarrow 0$ and $\infty$ . Find $r$ that minimizes $V(r)$ . Discuss changes in energy interaction with $A=4$ .

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Problem 13817

Given $\frac{\mathrm{dN}}{\mathrm{dt}}=\mathrm{rN}$ , find the per capita growth rate and whether $\mathrm{N}(1)$ is > or < 20 if $\mathrm{r}<0$ and $\mathrm{N}(0)=20$ .

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Problem 13818

Graph the function $g(x)=\left|x^{2}-3\right|$ and identify where it's not differentiable. Provide exact values.

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Problem 13819

What lot size $x$ minimizes total inventory costs given $C(x)=2.5 x+90000 x^{-1}+30000$ ? Find the minimum cost in \$.

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Problem 13820

Approximate the area under $y=x^{2}$ from $x=0$ to $x=3$ using a Right Endpoint method with 6 subdivisions.

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Problem 13821

What lot size minimizes Glorious Gadgets' inventory costs given $C(x)=2.5x+90000x^{-1}+30000$ ? What's the minimum cost?

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Problem 13822

Find the area represented by the definite integral $\int_{2}^{6}(4 x-16) d x$ .

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Problem 13823

A train's position is given by $s(t)=\frac{60}{t}$ for $3 \leq t \leq 8$ .
(a) Graph $s(t)$ . (b) Find the average velocity from $t=3$ to $t=8$ . Average velocity: $\square$ (round to nearest tenth).

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Problem 13824

Find the derivative of $f(x)=2^{3 x^{2}}$ .

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Problem 13825

Calculate the average value of $g(t)=3^{t}+4$ on $[1,6]$ and round to the nearest hundredth.

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Problem 13826

Calculate $\int_{C} f d s$ for the curve where $f(x, y, z) = 2x^{2} + 72z$ and $\mathbf{c}(t) = (e^{t}, 3t^{2}, t)$ for $0 \leq t \leq 1$ .

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Problem 13827

Write the left Riemann sum for $f(x)=\frac{3}{x}$ on $[1,6]$ with $n=30$ and evaluate it. Choose A, B, C, or D.

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Problem 13828

Find the average value of a yacht over its first 6 years, given $V(t)=250000 e^{-0.11 t}$ . $V_{\text{ave}}=\$$

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Problem 13829

Evaluate the integral $\int_{C} x y^{4} d s$ for the right half of the circle $x^{2}+y^{2}=9$ in clockwise direction.

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Problem 13830

Expand the function $f(z)=\frac{1}{10-5z}$ in a Maclaurin series and find the radius of convergence $R$ .

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Problem 13831

Find the derivative of $f(x)=\left[\sin \left(4 e^{2 x}+1\right)\right]^{2}$ .

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Problem 13832

Evaluate the line integral $\int_{C} x y d x + y d y$ for $x=8t, y=14t$ , where $0 \leq t \leq 1$ .

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Problem 13833

Calculate the work of the vector field $\mathbf{F}(x, y, z)=x \mathbf{i}+9 x y \mathbf{j}-(x+z) \mathbf{k}$ from $(1,4,2)$ to $(0,5,1)$ .

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Problem 13834

Calculate the line integral $\int_{C} f d s$ where $f(x, y, z)=2 x^{2}+72 z$ and $\mathbf{c}(t)=\left(e^{t}, 3 t^{2}, t\right)$ for $0 \leq t \leq 1$ .

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Problem 13835

Evaluate the integral $\int_{C} y^{2} dx + (xy - x^{2}) dy$ along the path $C: y=9x$ from $(0,0)$ to $(1,9)$ .

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Problem 13836

Find $y$ using implicit differentiation from $y^{\prime \prime} = x^{2} + 4y^{2}$ or $x^{2} + 4y^{2} = 4$ .

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Problem 13837

Differentiate the function $y=\log_{4}(e^{-x} \cos(\pi x))$ .

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Problem 13838

Find the average temperature of coffee over the first 30 minutes given $T(t)=20+75 e^{-t / 50}$ .

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Problem 13839

If $f(x)$ has a local minimum at $x=c$ with $f^{\prime}(x)<0$ for $x<c$ and $f^{\prime}(x)>0$ for $x>c$ , then $x=c$ is a: a) absolute maximum b) local minimum c) absolute minimum d) local maximum

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Problem 13840

Determine where the function $f(x)=\frac{x^{4}}{4}-3 x^{3}-1$ is concave down: $(-\infty, 0)$ , $(-\infty, 6)$ , $(0,6)$ , $(6, \infty)$ , or never.

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Problem 13841

Determine if $x=1$ is a local minimum, local maximum, absolute minimum, or absolute maximum for $f$ given $f^{\prime \prime}(x)=2x-1$ .

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Problem 13842

Bestimmen Sie den Differenzenquotienten von $f(x)=\frac{1}{2} x^{2}-4$ für die Intervalle: a) $I=[0 ; 2]$ , b) $I=[-1 ; 1]$ , c) $I=[-1 ; 2]$ , d) $I=[-2 ; 1]$ .

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Problem 13843

Find the line integral of $5 y^{2} \vec{i}+3 x \vec{j}$ along the segment from $(6,2)$ to $(0,0)$ .

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Problem 13844

Find the local min and max of $g(x)=x \sqrt{9-x^{2}}$ .

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Problem 13845

Dans un repère $(0, \vec{\imath}, \vec{\jmath}, \vec{k})$ , le vecteur-position est $\overrightarrow{O M}=3 t \vec{\imath}+(-4 t^{2}+2 t+3) \vec{\jmath}+5 \vec{k}$ .
1) Équations horaires du mouvement. 2) Coordonnées du vecteur-vitesse. 3) Norme du vecteur-vitesse. 4) Coordonnées du vecteur-accélération.

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Problem 13846

Find the partial derivative of $x^{3}-x y+y^{3}=23$ with respect to $x$ .

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Problem 13847

Find the derivatives of these functions: a) $f(x)=\frac{x^{2}+1}{x-1}$ b) $g(x)=\left(x^{2}+3x-5\right) \frac{2}{3}$

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Problem 13848

Find the derivative of $f(x)=\csc(2x)\sqrt{4x^{2}+3}$ .

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Problem 13849

Compute the line integral $\int_{C} f d s$ where $f(x, y, z) = 2x^{2} + 72z$ and $\mathbf{c}(t) = (e^{t}, 3t^{2}, t)$ for $0 \leq t \leq 1$ .

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Problem 13850

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

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Problem 13851

Evaluate $f(x)=16-x^{2/3}$ at $x=-64$ and $x=64$ . Find $c$ in $(-64,64)$ where $f'(c)=0$ . Discuss Rolle's Theorem.

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Problem 13852

Expand $f(z)=\frac{1}{2-z}$ in a Taylor series at $z_{0}=4i$ and find the radius of convergence $R$ .

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Problem 13853

Find the tangent line equation at the point (1,0) for the curve $x^{2}-x y-y^{2}=1$ .

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Problem 13854

Find the tangent line to the curve $y \sin(12x) = x \cos(2y)$ at the point $(\frac{\pi}{2}, \frac{\pi}{4})$ using implicit differentiation.

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Problem 13855

Find the tangent line equation to $y \sin (12 x) = x \cos (2 y)$ at $(\pi / 2, \pi / 4)$ .

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Problem 13856

Find the tangent line equation for $y=16 x-\frac{12}{x^{2}}+2$ at $x=2$ .

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Problem 13857

Coût de production d' $x$ PlayStations 6: $C(x)=800,000+340x+0.0005x^{2}$ . Trouvez $x$ pour minimiser le coût moyen.

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Problem 13858

Rewrite the integral $\int_{2}^{6}(5 x^{3}-6 x) dx$ as the difference of two integrals. Choose the correct option.

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Problem 13859

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

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Problem 13860

Given the function $f(x)=x^{3}-3 x+5$ on $[-2,2]$ , is $f$ continuous there? Find $f^{\prime}(x)$ , $f(-2)$ , $f(2)$ , and $\frac{f(b)-f(a)}{b-a}$ . Can the mean value theorem apply?

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Problem 13861

Explain why $\int_{a}^{a} f(x) d x=0$ . Choose the correct answer: A, B, C, or D.

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Problem 13862

Find the derivative of $y=(\ln (x))^{\cos (6 x)}$ using logarithmic differentiation.

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Problem 13863

Find local maxima, minima, and intervals of increase/decrease for $f^{\prime}(x)=(x-1)(x+1)(x+4)$ . Sketch a possible graph.

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Problem 13864

Find the derivative of $f(x)=\sqrt{x}(\sqrt{x}-2)$ .

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Problem 13865

Find local extrema and inflection points for functions: $f(x)=x^{3}-6x^{2}+5$ , $g(x)=x\sqrt{9-x^{2}}$ , $h(x)=xe^{-\frac{x^{2}}{9}}$ , $j(x)=\ln(x^{4}+27)$ , $k(x)=x\sqrt{6-x}$ , $l(x)=\sqrt[3]{x^{2}-1}$ . Also, calculate elasticity for $q=500-10p$ at $p=\$30$ .

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Problem 13866

Given the function $f(x)=x^{3}-4 x^{2}-16 x+6$ on $[-4,4]$ , is $f$ continuous? Find $f^{\prime}(x)$ and $f(-4)$ , $f(4)$ . Can Rolle's theorem apply?

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Problem 13867

Find points on the curve $y=2-\frac{1}{x}$ where the tangent slope equals $\frac{1}{4}$ .

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Problem 13868

Find the tangent line to the curve $y \sin (12 x) = x \cos (2 y)$ at the point $(\pi / 2, \pi / 4)$ .

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Problem 13869

Estimate the area under $f(x)=-4x-5$ from $x=0$ to $x=4$ using a left Riemann sum.

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Problem 13870

Prove that $x^{3}-14 x+c=0$ has at most one solution in $[-2,2]$ using Rolle's theorem.

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Problem 13871

Déterminez les limites $\lim _{x \rightarrow-\infty} f(x)$ et $\lim _{x \rightarrow \infty} f(x)$ pour $f(x)=x e^{1-2 x^{2}}$ .

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Problem 13872

Coût de production pour $x$ PlayStations 6 : $C(x)=800,000+340 x+0.0005 x^{2}$ . Trouver $x$ pour coût moyen minimal et calculer ce coût.

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Problem 13873

Evaluate the integral $\int_{0}^{4}(x^{2}-4) dx=\square($ Simplify your answer.)

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Problem 13874

Evaluate the integral $\int_{0}^{4}(x^{2}-4) dx$ using the Riemann Sum. Choose the correct option from A, B, C, or D.

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Problem 13875

Evaluate the integral $2 \pi \int_{15}^{15.8}\left(\sqrt{0.8^{2}-(x-15.8)^{2}}\right) \times \sqrt{1+\left[\frac{-5 \sqrt{5} x+79 \sqrt{5}}{5 \sqrt{-5 x^{2}+158 x-1245}}\right]^{2}} dx$ using numerical methods.

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Problem 13876

Evaluate the integral $\int_{3}^{4} 9 f(x) d x$ given $\int_{3}^{7} f(x) d x=14$ and $\int_{4}^{7} f(x) d x=8$ .

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Problem 13877

Find the derivative of the function $\frac{-e^{x}}{3 y^{2}}$ with respect to 'x'.

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Problem 13878

Evaluate the integrals given the functions $f$ and $g$ on $[3,7]$ : a. $\int_{3}^{4} 9 f(x) d x$ b. $\int_{3}^{7}(f(x)-g(x)) d x$

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Problem 13879

Evaluate the integrals based on given functions $f$ and $g$ on $[3,7]$ with specific integral values. a. $\int_{3}^{4} 9 f(x) d x$ b. $\int_{3}^{7}(f(x)-g(x)) d x$ c. $\int_{3}^{4}(f(x)-g(x)) d x$

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Problem 13880

Find the derivative of $f(y) = \frac{-e^{x}}{3 y^{2}}$ with respect to $y$ .

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Problem 13881

Evaluate the following integrals given $\int_{3}^{7} f(x) d x=14$ , $\int_{3}^{7} g(x) d x=7$ :
a. $\int_{3}^{4} 9 f(x) d x$ b. $\int_{3}^{7}(f(x)-g(x)) d x$ c. $\int_{3}^{4}(f(x)-g(x)) d x$ d. $\int_{4}^{7}(g(x)-f(x)) d x$ e. $\int_{4}^{7} 6 g(x) d x$

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Problem 13882

Evaluate $\int_{5}^{1} 6 x(8-x) d x$ using $\int_{1}^{5} 6 x(8-x) d x=328$ . Choose A or B.

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Problem 13883

Evaluate the integrals given the functions $f$ and $g$ on $[3,7]$ with the following properties:
a. $\int_{3}^{4} 9 f(x) d x$ b. $\int_{3}^{7}(f(x)-g(x)) d x$ c. $\int_{3}^{4}(f(x)-g(x)) d x$ d. $\int_{4}^{7}(g(x)-f(x)) d x$

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Problem 13884

Evaluate the integrals using $\int_{1}^{5} 6 x(8-x) d x=328$ .
a. Find $\int_{5}^{1} 6 x(8-x) d x$ . b. Find $\int_{1}^{5} x(8-x) d x$ .

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Problem 13885

Evaluate the following integrals given $\int_{3}^{7} f(x) d x=14$ , $\int_{3}^{7} g(x) d x=7$ , $\int_{4}^{7} f(x) d x=8$ , and $\int_{3}^{4} g(x) d x=3$ :
a. $\int_{3}^{4} 9 f(x) d x$ b. $\int_{3}^{7}(f(x)-g(x)) d x$ c. $\int_{3}^{4}(f(x)-g(x)) d x$ d. $\int_{4}^{7}(g(x)-f(x)) d x$ e. $\int_{4}^{7} 6 g(x) d x$ f. $\int_{4}^{3} 4 f(x) d x=\square$

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Problem 13886

Find the absolute extreme values of $f(x)=(x+3)^{\frac{4}{3}}$ on $[-5,5]$ . Max: A/B, Min: A/B.

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Problem 13887

Find the radius of convergence for the series $\sum(c_{n}+d_{n}) x^{n}$ given that $\sum c_{n} x^{n}$ has radius 2 and $\sum d_{n} x^{n}$ has radius 3.

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Problem 13888

Determine the local max and min of $f$ using First and Second Derivative Tests: $f(x)=8+9x^{2}-6x^{3}$ .

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Problem 13889

Evaluate $\int_{7}^{2} g(x) d x$ given $\int_{2}^{7} g(x) d x=8$ .

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Problem 13890

Calculate the integral $\int \frac{x-5}{(2 x-3)(x-1)} d x$ .

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Problem 13891

Given the function $f(x)=e^{-2 / x}$ , find:
(a) Vertical asymptote(s): $x=$ ; Horizontal asymptote(s): $y=$ . (b) Intervals of increase and decrease in interval notation. (c) Local max and min values. (d) Intervals of concavity: up and down, plus the inflection point $(x, y)=(\square)$ .

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Problem 13892

Given $\int_{2}^{4} f(x) d x=-4$ , $\int_{2}^{7} f(x) d x=4$ , and $\int_{2}^{7} g(x) d x=8$ , find:
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 3 g(x) d x$
3. $\int_{2}^{7}[g(x)-f(x)] d x=\square$

See Solution

Problem 13893

Evaluate the following integrals using $\int_{1}^{5} 6 x(8-x) d x=328$ :
A. $\int_{1} x(8-x) d x$ B. The integral cannot be calculated. C. $\int_{5}^{1} 12 x(8-x) d x$ D. $\int_{1}^{9} 6 x(8-x) d x=\square$ E. The integral cannot be calculated.

See Solution

Problem 13894

Analyze the series $\Sigma a_{n}$ for these limits: (a) 8, (b) 0.8, (c) 1. What can you conclude?

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Problem 13895

Evaluate the following integrals using $\int_{1}^{5} 6 x(8-x) d x=328$ :
A. $\int_{5} 6 x(8-x) d x$ B. $\int_{1}^{5} x(8-x) d x$ C. $\int_{5}^{1} 12 x(8-x) d x$

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Problem 13896

Find the derivative of $f(x)=x^{x} \cos (x)$ using logarithmic differentiation.

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Problem 13897

Given $\int_{2}^{4} f(x) d x=-4$ , $\int_{2}^{4} f(x) d x=4$ , $\int_{2}^{6} g(x) d x=8$ , find:
1. $\int_{7}^{2} g(x) d x$ .
2. $\int_{2}^{7} 3 g(x) d x$ .
3. $\int_{2}^{7}[g(x)-f(x)] d x$ .
4. $\int_{2}^{7}[3 g(x)-f(x)] d x=\square$ .

See Solution

Problem 13898

Given $\int_{2}^{4} f(x) d x=-4$ , $\int_{2}^{7} f(x) d x=4$ , and $\int_{2}^{7} g(x) d x=8$ , find:
1. $\int_{7}^{2} g(x) d x$ .
2. $\int_{2}^{7} 3 g(x) d x$ .
3. $\int_{2}^{7}[g(x)-f(x)] d x$ .
4. $\int_{2}^{7}[3 g(x)-f(x)] d x=\square$ .

See Solution

Problem 13899

Given $\int_{2}^{4} f(x) d x=-4$ , $\int_{2}^{7} f(x) d x=4$ , and $\int_{2}^{7} g(x) d x=8$ , find:
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 3 g(x) d x=\square$

See Solution

Problem 13900

Sketch the graph of $f(x)=-25+5x$ on $[0,20]$ and find the net area using left, right, and midpoint Riemann sums with $n=4$ .

See Solution

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Calculus

Problem 13801

Find the dimensions of a rectangular tank with a square base and a volume of 37,044ft337,044 \mathrm{ft}^{3}37,044ft3 that minimizes surface area. The height is □ft\square \mathrm{ft}□ft and the base side length is □ft\square \mathrm{ft}□ft.

Problem 13802

Analyze the Leonard-Jones 6-3 potential: V(r)=1r6−Ar3V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}V(r)=r61​−r3A​. What happens as r→0r \rightarrow 0r→0?

Problem 13803

Analyze the function V(r)=1r6−Ar3V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}V(r)=r61​−r3A​ for r>0r>0r>0. (a) Determine the limits as r→0r \rightarrow 0r→0 and r→∞r \rightarrow \inftyr→∞. (b) Discuss the existence of a minimum for V(r)V(r)V(r).

Problem 13804

Given the function f(x)=x2−1f(x)=x^{2}-1f(x)=x2−1, find the derivative at x=2x=2x=2, f(2)f(2)f(2), tangent line equation, and graph both.

Problem 13805

Use the function f(x)=x2−1f(x)=x^{2}-1f(x)=x2−1 to find: (a) the derivative at x=2x=2x=2, (b) f(2)f(2)f(2) and tangent line, (c) graph both.

Problem 13806

Analyze the function V(r)=1r6−Ar3V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}V(r)=r61​−r3A​ for r>0r>0r>0. (a) What happens to V(r)V(r)V(r) as r→0r \rightarrow 0r→0 and r→∞r \rightarrow \inftyr→∞? (b) Explain if a minimum exists and find that minimum value of rrr. r=□r=\squarer=□

Problem 13807

What is the limit of V(r)=e−r−Ae−arV(r)=e^{-r}-A e^{-a r}V(r)=e−r−Ae−ar as r→0r \rightarrow 0r→0 for a=12a=\frac{1}{2}a=21​ and A=1A=1A=1?

Problem 13808

Analyze the function V(r)=1r6−Ar3V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}V(r)=r61​−r3A​ for r>0r>0r>0. What happens as r→0r \to 0r→0 and r→∞r \to \inftyr→∞?

Problem 13809

Find conditions for a polynomial to have an absolute minimum. Analyze a graph to find its lowest degree and concave down interval.

Problem 13810

Fish interactions are modeled by the Morse potential V(r)=e−r−Ae−arV(r)=e^{-r}-A e^{-a r}V(r)=e−r−Ae−ar. With a=12a=\frac{1}{2}a=21​ and A=1A=1A=1, analyze V(r)V(r)V(r) as r→0r \rightarrow 0r→0 and ∞\infty∞, then find rrr that minimizes V(r)V(r)V(r). r=□r=\squarer=□

Problem 13811

Given the function f(x)=x2+1f(x)=x^{2}+1f(x)=x2+1, find the derivative at x=2x=2x=2, f(2)f(2)f(2), the tangent line, and graph them.

Problem 13812

Analyze the function V(r)=1r6−Ar3V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}V(r)=r61​−r3A​ for behavior as r→∞r \to \inftyr→∞, find minimizing rrr, and consider A<0A<0A<0.

Problem 13813

Analyze the function V(r)=e−r−Ae−arV(r)=e^{-r}-A e^{-a r}V(r)=e−r−Ae−ar for a=12a=\frac{1}{2}a=21​, A=1A=1A=1: behavior as r→0r \to 0r→0 and r→∞r \to \inftyr→∞. Find rrr that minimizes V(r)V(r)V(r).

Problem 13814

Analyze the function V(r)=e−r−Ae−arV(r)=e^{-r}-A e^{-a r}V(r)=e−r−Ae−ar for a=12a=\frac{1}{2}a=21​ and A=1A=1A=1. What happens as r→0r \rightarrow 0r→0 and r→∞r \rightarrow \inftyr→∞?

Problem 13815

Analyze the function V(r)=e−r−Ae−arV(r)=e^{-r}-A e^{-a r}V(r)=e−r−Ae−ar for a=12a=\frac{1}{2}a=21​, A=1A=1A=1 and find its minimum value for r>0r>0r>0.

Problem 13816

Given the function V(r)=e−r−Ae−12rV(r)=e^{-r}-Ae^{-\frac{1}{2}r}V(r)=e−r−Ae−21​r for r>0r>0r>0, analyze its behavior as r→0r \rightarrow 0r→0 and ∞\infty∞. Find rrr that minimizes V(r)V(r)V(r). Discuss changes in energy interaction with A=4A=4A=4.

Problem 13817

Given dNdt=rN\frac{\mathrm{dN}}{\mathrm{dt}}=\mathrm{rN}dtdN​=rN, find the per capita growth rate and whether N(1)\mathrm{N}(1)N(1) is > or < 20 if r<0\mathrm{r}<0r<0 and N(0)=20\mathrm{N}(0)=20N(0)=20.

Problem 13818

Graph the function g(x)=∣x2−3∣g(x)=\left|x^{2}-3\right|g(x)=∣∣​x2−3∣∣​ and identify where it's not differentiable. Provide exact values.

Problem 13819

What lot size xxx minimizes total inventory costs given C(x)=2.5x+90000x−1+30000C(x)=2.5 x+90000 x^{-1}+30000C(x)=2.5x+90000x−1+30000? Find the minimum cost in \$.

Problem 13820

Approximate the area under y=x2y=x^{2}y=x2 from x=0x=0x=0 to x=3x=3x=3 using a Right Endpoint method with 6 subdivisions.

Problem 13821

What lot size minimizes Glorious Gadgets' inventory costs given C(x)=2.5x+90000x−1+30000C(x)=2.5x+90000x^{-1}+30000C(x)=2.5x+90000x−1+30000? What's the minimum cost?

Problem 13822

Find the area represented by the definite integral ∫26(4x−16)dx\int_{2}^{6}(4 x-16) d x∫26​(4x−16)dx.

Problem 13823

A train's position is given by s(t)=60ts(t)=\frac{60}{t}s(t)=t60​ for 3≤t≤83 \leq t \leq 83≤t≤8. (a) Graph s(t)s(t)s(t). (b) Find the average velocity from t=3t=3t=3 to t=8t=8t=8. Average velocity: □\square□ (round to nearest tenth).

Problem 13824

Find the derivative of f(x)=23x2f(x)=2^{3 x^{2}}f(x)=23x2.

Problem 13825

Calculate the average value of g(t)=3t+4g(t)=3^{t}+4g(t)=3t+4 on [1,6][1,6][1,6] and round to the nearest hundredth.

Problem 13826

Calculate ∫Cfds\int_{C} f d s∫C​fds for the curve where f(x,y,z)=2x2+72zf(x, y, z) = 2x^{2} + 72zf(x,y,z)=2x2+72z and c(t)=(et,3t2,t)\mathbf{c}(t) = (e^{t}, 3t^{2}, t)c(t)=(et,3t2,t) for 0≤t≤10 \leq t \leq 10≤t≤1.

Problem 13827

Write the left Riemann sum for f(x)=3xf(x)=\frac{3}{x}f(x)=x3​ on [1,6][1,6][1,6] with n=30n=30n=30 and evaluate it. Choose A, B, C, or D.

Problem 13828

Find the average value of a yacht over its first 6 years, given V(t)=250000e−0.11tV(t)=250000 e^{-0.11 t}V(t)=250000e−0.11t. Vave=$V_{\text{ave}}=\$Vave​=$

Problem 13829

Evaluate the integral ∫Cxy4ds\int_{C} x y^{4} d s∫C​xy4ds for the right half of the circle x2+y2=9x^{2}+y^{2}=9x2+y2=9 in clockwise direction.

Problem 13830

Expand the function f(z)=110−5zf(z)=\frac{1}{10-5z}f(z)=10−5z1​ in a Maclaurin series and find the radius of convergence RRR.

Problem 13831

Find the derivative of f(x)=[sin⁡(4e2x+1)]2f(x)=\left[\sin \left(4 e^{2 x}+1\right)\right]^{2}f(x)=[sin(4e2x+1)]2.

Problem 13832

Evaluate the line integral ∫Cxydx+ydy\int_{C} x y d x + y d y∫C​xydx+ydy for x=8t,y=14tx=8t, y=14tx=8t,y=14t, where 0≤t≤10 \leq t \leq 10≤t≤1.

Problem 13833

Calculate the work of the vector field F(x,y,z)=xi+9xyj−(x+z)k\mathbf{F}(x, y, z)=x \mathbf{i}+9 x y \mathbf{j}-(x+z) \mathbf{k}F(x,y,z)=xi+9xyj−(x+z)k from (1,4,2)(1,4,2)(1,4,2) to (0,5,1)(0,5,1)(0,5,1).

Problem 13834

Calculate the line integral ∫Cfds\int_{C} f d s∫C​fds where f(x,y,z)=2x2+72zf(x, y, z)=2 x^{2}+72 zf(x,y,z)=2x2+72z and c(t)=(et,3t2,t)\mathbf{c}(t)=\left(e^{t}, 3 t^{2}, t\right)c(t)=(et,3t2,t) for 0≤t≤10 \leq t \leq 10≤t≤1.

Problem 13835

Evaluate the integral ∫Cy2dx+(xy−x2)dy\int_{C} y^{2} dx + (xy - x^{2}) dy∫C​y2dx+(xy−x2)dy along the path C:y=9xC: y=9xC:y=9x from (0,0)(0,0)(0,0) to (1,9)(1,9)(1,9).

Problem 13836

Find yyy using implicit differentiation from y′′=x2+4y2y^{\prime \prime} = x^{2} + 4y^{2}y′′=x2+4y2 or x2+4y2=4x^{2} + 4y^{2} = 4x2+4y2=4.

Problem 13837

Differentiate the function y=log⁡4(e−xcos⁡(πx))y=\log_{4}(e^{-x} \cos(\pi x))y=log4​(e−xcos(πx)).

Problem 13838

Find the average temperature of coffee over the first 30 minutes given T(t)=20+75e−t/50T(t)=20+75 e^{-t / 50}T(t)=20+75e−t/50.

Problem 13839

If f(x)f(x)f(x) has a local minimum at x=cx=cx=c with f′(x)<0f^{\prime}(x)<0f′(x)<0 for x<cx<cx<c and f′(x)>0f^{\prime}(x)>0f′(x)>0 for x>cx>cx>c, then x=cx=cx=c is a: a) absolute maximum b) local minimum c) absolute minimum d) local maximum

Problem 13840

Find the dimensions of a rectangular tank with a square base and a volume of $37,044 \mathrm{ft}^{3}$ that minimizes surface area. The height is $\square \mathrm{ft}$ and the base side length is $\square \mathrm{ft}$ .

Analyze the Leonard-Jones 6-3 potential: $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ . What happens as $r \rightarrow 0$ ?

Analyze the function $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ for $r>0$ .
(a) Determine the limits as $r \rightarrow 0$ and $r \rightarrow \infty$ .
(b) Discuss the existence of a minimum for $V(r)$ .

Given the function $f(x)=x^{2}-1$ , find the derivative at $x=2$ , $f(2)$ , tangent line equation, and graph both.

Use the function $f(x)=x^{2}-1$ to find: (a) the derivative at $x=2$ , (b) $f(2)$ and tangent line, (c) graph both.

Analyze the function $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ for $r>0$ .
(a) What happens to $V(r)$ as $r \rightarrow 0$ and $r \rightarrow \infty$ ? (b) Explain if a minimum exists and find that minimum value of $r$ . $r=\square$

What is the limit of $V(r)=e^{-r}-A e^{-a r}$ as $r \rightarrow 0$ for $a=\frac{1}{2}$ and $A=1$ ?

Analyze the function $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ for $r>0$ . What happens as $r \to 0$ and $r \to \infty$ ?

Fish interactions are modeled by the Morse potential $V(r)=e^{-r}-A e^{-a r}$ . With $a=\frac{1}{2}$ and $A=1$ , analyze $V(r)$ as $r \rightarrow 0$ and $\infty$ , then find $r$ that minimizes $V(r)$ . $r=\square$

Given the function $f(x)=x^{2}+1$ , find the derivative at $x=2$ , $f(2)$ , the tangent line, and graph them.

Analyze the function $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ for behavior as $r \to \infty$ , find minimizing $r$ , and consider $A<0$ .

Analyze the function $V(r)=e^{-r}-A e^{-a r}$ for $a=\frac{1}{2}$ , $A=1$ : behavior as $r \to 0$ and $r \to \infty$ . Find $r$ that minimizes $V(r)$ .

Analyze the function $V(r)=e^{-r}-A e^{-a r}$ for $a=\frac{1}{2}$ and $A=1$ . What happens as $r \rightarrow 0$ and $r \rightarrow \infty$ ?

Analyze the function $V(r)=e^{-r}-A e^{-a r}$ for $a=\frac{1}{2}$ , $A=1$ and find its minimum value for $r>0$ .

Given the function $V(r)=e^{-r}-Ae^{-\frac{1}{2}r}$ for $r>0$ , analyze its behavior as $r \rightarrow 0$ and $\infty$ . Find $r$ that minimizes $V(r)$ . Discuss changes in energy interaction with $A=4$ .

Given $\frac{\mathrm{dN}}{\mathrm{dt}}=\mathrm{rN}$ , find the per capita growth rate and whether $\mathrm{N}(1)$ is > or < 20 if $\mathrm{r}<0$ and $\mathrm{N}(0)=20$ .

Graph the function $g(x)=\left|x^{2}-3\right|$ and identify where it's not differentiable. Provide exact values.

What lot size $x$ minimizes total inventory costs given $C(x)=2.5 x+90000 x^{-1}+30000$ ? Find the minimum cost in \$.

Approximate the area under $y=x^{2}$ from $x=0$ to $x=3$ using a Right Endpoint method with 6 subdivisions.

What lot size minimizes Glorious Gadgets' inventory costs given $C(x)=2.5x+90000x^{-1}+30000$ ? What's the minimum cost?

Find the area represented by the definite integral $\int_{2}^{6}(4 x-16) d x$ .

A train's position is given by $s(t)=\frac{60}{t}$ for $3 \leq t \leq 8$ .
(a) Graph $s(t)$ . (b) Find the average velocity from $t=3$ to $t=8$ . Average velocity: $\square$ (round to nearest tenth).

Find the derivative of $f(x)=2^{3 x^{2}}$ .

Calculate the average value of $g(t)=3^{t}+4$ on $[1,6]$ and round to the nearest hundredth.

Calculate $\int_{C} f d s$ for the curve where $f(x, y, z) = 2x^{2} + 72z$ and $\mathbf{c}(t) = (e^{t}, 3t^{2}, t)$ for $0 \leq t \leq 1$ .

Write the left Riemann sum for $f(x)=\frac{3}{x}$ on $[1,6]$ with $n=30$ and evaluate it. Choose A, B, C, or D.

Find the average value of a yacht over its first 6 years, given $V(t)=250000 e^{-0.11 t}$ . $V_{\text{ave}}=\$$

Evaluate the integral $\int_{C} x y^{4} d s$ for the right half of the circle $x^{2}+y^{2}=9$ in clockwise direction.

Expand the function $f(z)=\frac{1}{10-5z}$ in a Maclaurin series and find the radius of convergence $R$ .

Find the derivative of $f(x)=\left[\sin \left(4 e^{2 x}+1\right)\right]^{2}$ .

Evaluate the line integral $\int_{C} x y d x + y d y$ for $x=8t, y=14t$ , where $0 \leq t \leq 1$ .

Calculate the work of the vector field $\mathbf{F}(x, y, z)=x \mathbf{i}+9 x y \mathbf{j}-(x+z) \mathbf{k}$ from $(1,4,2)$ to $(0,5,1)$ .

Calculate the line integral $\int_{C} f d s$ where $f(x, y, z)=2 x^{2}+72 z$ and $\mathbf{c}(t)=\left(e^{t}, 3 t^{2}, t\right)$ for $0 \leq t \leq 1$ .

Evaluate the integral $\int_{C} y^{2} dx + (xy - x^{2}) dy$ along the path $C: y=9x$ from $(0,0)$ to $(1,9)$ .

Find $y$ using implicit differentiation from $y^{\prime \prime} = x^{2} + 4y^{2}$ or $x^{2} + 4y^{2} = 4$ .

Differentiate the function $y=\log_{4}(e^{-x} \cos(\pi x))$ .

Find the average temperature of coffee over the first 30 minutes given $T(t)=20+75 e^{-t / 50}$ .

If $f(x)$ has a local minimum at $x=c$ with $f^{\prime}(x)<0$ for $x<c$ and $f^{\prime}(x)>0$ for $x>c$ , then $x=c$ is a: a) absolute maximum b) local minimum c) absolute minimum d) local maximum

Determine where the function $f(x)=\frac{x^{4}}{4}-3 x^{3}-1$ is concave down: $(-\infty, 0)$ , $(-\infty, 6)$ , $(0,6)$ , $(6, \infty)$ , or never.

Determine if $x=1$ is a local minimum, local maximum, absolute minimum, or absolute maximum for $f$ given $f^{\prime \prime}(x)=2x-1$ .

Bestimmen Sie den Differenzenquotienten von $f(x)=\frac{1}{2} x^{2}-4$ für die Intervalle: a) $I=[0 ; 2]$ , b) $I=[-1 ; 1]$ , c) $I=[-1 ; 2]$ , d) $I=[-2 ; 1]$ .

Find the line integral of $5 y^{2} \vec{i}+3 x \vec{j}$ along the segment from $(6,2)$ to $(0,0)$ .

Find the local min and max of $g(x)=x \sqrt{9-x^{2}}$ .

Find the partial derivative of $x^{3}-x y+y^{3}=23$ with respect to $x$ .

Find the derivatives of these functions: a) $f(x)=\frac{x^{2}+1}{x-1}$ b) $g(x)=\left(x^{2}+3x-5\right) \frac{2}{3}$

Find the derivative of $f(x)=\csc(2x)\sqrt{4x^{2}+3}$ .

Compute the line integral $\int_{C} f d s$ where $f(x, y, z) = 2x^{2} + 72z$ and $\mathbf{c}(t) = (e^{t}, 3t^{2}, t)$ for $0 \leq t \leq 1$ .

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

Evaluate $f(x)=16-x^{2/3}$ at $x=-64$ and $x=64$ . Find $c$ in $(-64,64)$ where $f'(c)=0$ . Discuss Rolle's Theorem.

Expand $f(z)=\frac{1}{2-z}$ in a Taylor series at $z_{0}=4i$ and find the radius of convergence $R$ .

Find the tangent line equation at the point (1,0) for the curve $x^{2}-x y-y^{2}=1$ .

Find the tangent line to the curve $y \sin(12x) = x \cos(2y)$ at the point $(\frac{\pi}{2}, \frac{\pi}{4})$ using implicit differentiation.

Find the tangent line equation to $y \sin (12 x) = x \cos (2 y)$ at $(\pi / 2, \pi / 4)$ .

Find the tangent line equation for $y=16 x-\frac{12}{x^{2}}+2$ at $x=2$ .

Coût de production d' $x$ PlayStations 6: $C(x)=800,000+340x+0.0005x^{2}$ . Trouvez $x$ pour minimiser le coût moyen.

Rewrite the integral $\int_{2}^{6}(5 x^{3}-6 x) dx$ as the difference of two integrals. Choose the correct option.

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

Given the function $f(x)=x^{3}-3 x+5$ on $[-2,2]$ , is $f$ continuous there? Find $f^{\prime}(x)$ , $f(-2)$ , $f(2)$ , and $\frac{f(b)-f(a)}{b-a}$ . Can the mean value theorem apply?

Explain why $\int_{a}^{a} f(x) d x=0$ . Choose the correct answer: A, B, C, or D.

Find the derivative of $y=(\ln (x))^{\cos (6 x)}$ using logarithmic differentiation.

Find local maxima, minima, and intervals of increase/decrease for $f^{\prime}(x)=(x-1)(x+1)(x+4)$ . Sketch a possible graph.

Find the derivative of $f(x)=\sqrt{x}(\sqrt{x}-2)$ .

Given the function $f(x)=x^{3}-4 x^{2}-16 x+6$ on $[-4,4]$ , is $f$ continuous? Find $f^{\prime}(x)$ and $f(-4)$ , $f(4)$ . Can Rolle's theorem apply?

Find points on the curve $y=2-\frac{1}{x}$ where the tangent slope equals $\frac{1}{4}$ .

Find the tangent line to the curve $y \sin (12 x) = x \cos (2 y)$ at the point $(\pi / 2, \pi / 4)$ .

Estimate the area under $f(x)=-4x-5$ from $x=0$ to $x=4$ using a left Riemann sum.

Prove that $x^{3}-14 x+c=0$ has at most one solution in $[-2,2]$ using Rolle's theorem.

Déterminez les limites $\lim _{x \rightarrow-\infty} f(x)$ et $\lim _{x \rightarrow \infty} f(x)$ pour $f(x)=x e^{1-2 x^{2}}$ .

Coût de production pour $x$ PlayStations 6 : $C(x)=800,000+340 x+0.0005 x^{2}$ . Trouver $x$ pour coût moyen minimal et calculer ce coût.

Evaluate the integral $\int_{0}^{4}(x^{2}-4) dx=\square($ Simplify your answer.)

Evaluate the integral $\int_{0}^{4}(x^{2}-4) dx$ using the Riemann Sum. Choose the correct option from A, B, C, or D.