Calculus

Problem 3601

Find the surface area of the graph $y=2 x^{2}$ from $x=2$ to $x=4$ when revolved around the $x$ -axis. Set up the integral.

See Solution

Problem 3602

A particle moves along the curve $\vec{r}(t)=\left(\cos ^{\wedge} 2 t,-\cos ^{\wedge} 2 t, t\right)$ . Find its max speed and when it occurs, plus the acceleration related to direction change.

See Solution

Problem 3603

Stilus-Stifte hat eine Kostenfunktion $K(x)=2 x^{3}-18 x^{2}+62 x+32$ . a) Erklären Sie den Graphen und den Wendepunkt. b) Bei \$50 pro 1000 Bleistifte, wo wird Gewinn erzielt und wo ist er maximal?

See Solution

Problem 3604

Find the infinite limits: 1. $\lim _{x \rightarrow 3^{+}} \frac{8}{x-3}$ ; I. $\lim _{x \rightarrow 3^{-}} \frac{8}{x-3}$ .

See Solution

Problem 3605

Calculate the integral: $\int \frac{x^{2}-1}{x^{2}+1} d x$

See Solution

Problem 3606

Evaluate the integral: $\int \frac{2}{x^{2}+1} \, dx$ .

See Solution

Problem 3607

Find the direction of $r^{\prime}$ for $r(t)=\langle 5t-2, 7t+1, t^{2} \rangle$ at $t_{0}=1$ . Choose the correct option.

See Solution

Problem 3608

Evaluate the limit as $h$ approaches 0: $\lim_{h \rightarrow 0} \frac{2(3+h)^{2}-18}{h}$ . Simplify first.

See Solution

Problem 3609

Given the piecewise function:
$f(x) = \begin{cases} 14 & \text{if } x > 9 \\ 1 & \text{if } x = 9 \\ -x + 14 & \text{if } -8 \leq x < 9 \\ 22 & \text{if } x < -8 \end{cases}$
Sketch the graph and find the limits:
1. $\lim_{x \to 9^{-}} f(x)$
2. $\lim_{x \to 9^{+}} f(x)$
3. $\lim_{x \to 9} f(x)$
4. $\lim_{x \to -8^{-}} f(x)$
5. $\lim_{x \to -8^{+}} f(x)$

See Solution

Problem 3610

Sketch the graph of the piecewise function $f(x)$ and find the limits as specified:
1. $\lim _{x \rightarrow 9^{-}} f(x)$
2. $\lim _{x \rightarrow 9^{+}} f(x)$
3. $\lim _{x \rightarrow 9} f(x)$
4. $\lim _{x \rightarrow -8^{-}} f(x)$
5. $\lim _{x \rightarrow -8^{+}} f(x)$
6. $\lim _{x \rightarrow -8} f(x)$

See Solution

Problem 3611

Calculate the integral: $\int \frac{x^{2}+1}{x^{2}-1} d x$

See Solution

Problem 3612

Bestimme die Ableitungen für die Funktionen: a) $f(x)=2 x+x^{3}$ , b) $f(x)=5 x$ , c) $f(x)=a x^{2}$ , d) $f(x)=a x^{n}$ , e) $f(x)=2 x^{2}+4 x$ , f) $f(x)=\frac{1}{2} x^{2}+5$ , g) $f(x)=2 x^{3}-3 x^{2}+2$ , h) $f(x)=a x^{3}+b x+c$ .

See Solution

Problem 3613

Calculate the integral: $\int \frac{1}{\sqrt{x^{2}+8 x+7}} d x$

See Solution

Problem 3614

Evaluate the integral $\int \frac{1}{v=9-4 x^{2}} d x$ .

See Solution

Problem 3615

Find the difference quotient, $\frac{f(a+h)-f(a)}{h}$ , for $f(x)=\sqrt{x^{2}-x}$ .

See Solution

Problem 3616

Evaluate the integral: $\int \frac{1}{\sqrt{49-4 x^{2}}} d x$

See Solution

Problem 3617

Find the integral of $\frac{1}{x(x+2)}$ with respect to $x$ .

See Solution

Problem 3618

Calculate the integral $\int \frac{x^{3}}{x^{2}+1} d x$ .

See Solution

Problem 3619

Bestimmen Sie die Ableitung von $f(x)=x^{3}+\sqrt{x}$ .

See Solution

Problem 3620

Bestimmen Sie die Extrempunkte der Funktion $f(x) = \frac{1}{2} x^{3}-a x$ .

See Solution

Problem 3621

Find the derivative using the product rule for: $p(y)=\left(y^{-1}+y^{-2}\right)\left(2 y^{-3}-7 y^{-4}\right)$ . What is $p^{\prime}(y)$ ?

See Solution

Problem 3622

Bestimmen Sie, welcher Graph die Integralfunktion $I_{0}: x \mapsto \int_{0}^{x} f(t) dt$ und welcher die Funktion $\mathrm{I}_{1}: \mathrm{x} \mapsto \int_{1}^{\mathrm{x}} \mathrm{h}(\mathrm{t}) dt$ darstellt. Begründen Sie Ihre Wahl.

See Solution

Problem 3623

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{\ln x}{\sqrt{x}}$ .

See Solution

Problem 3624

Find $h^{\prime}(8)$ for $h(x)=\frac{f(x)}{g(x)}$ given $f(8)=1$ , $f'(8)=2$ , $g(8)=9$ , $g'(8)=6$ .

See Solution

Problem 3625

Find the tangent line equation to $f(x)=(3 x-2)(x+6)$ at the point $(1,7)$ .

See Solution

Problem 3626

Find the tangent line equation for $f(x)=(3x-5)(x+6)$ at the point $(2,8)$ .

See Solution

Problem 3627

Find the derivative of the function $f(x)=\frac{x^{3}+8}{4}$ .

See Solution

Problem 3628

5 Forscher untersuchen das Bakterienwachstum mit $A(t)=-0,005 t^{3}+0,2 t^{2}+0,9 t+1$ .
a) Fläche um 3 Uhr morgens? b) Maximale Zunahme der Fläche?

See Solution

Problem 3629

Find the values of $x$ where $f'(x)=0$ for the function $f(x)=(x^{2}-5)(x^{2}-\sqrt{5})$ .

See Solution

Problem 3630

Calculate the derivative of the product $f(x) g(x)$ using $f'(x) g'(x)$ for $f(x)=x^{2}+b x$ and $g(x)=c x+d$ . What is $f'(x) g'(x)$ ?

See Solution

Problem 3631

Calculate the derivatives:
(a) Find $f^{\prime}(x) g^{\prime}(x)$ for $f(x)=x^{2}+bx$ and $g(x)=cx+d$ .
(b) Compute $[f(x) g(x)]^{\prime}$ and show it differs from part (a).

See Solution

Problem 3632

Find the derivative of the function $f(x)=x^{9}$ .

See Solution

Problem 3633

Bestimmen Sie die Ableitung von $g(x)=x^{2 n}$ .

See Solution

Problem 3634

A bike company finds that a new employee assembles $M(d)=\frac{97 d^{2}}{4 d^{2}+11}$ bikes/day after $d$ days.
(a) Determine $M'(d)$ . (b) Calculate and explain $M'(2)$ and $M'(5)$ .

See Solution

Problem 3635

Find the derivative of $f(x)=x^{7}$ at the point $x_{0}=-1$ .

See Solution

Problem 3636

Find the derivative of the function $g(x)=x^{-2}$ using the power rule.

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Problem 3637

Find the derivative $s'(x)$ of the function $s(x) = \frac{x}{m+nx}$ and evaluate it at $x=50$ , $m=15$ , $n=3$ .

See Solution

Problem 3638

Rewrite $f(x)=\frac{1}{x^{7}}$ with $x$ in the numerator, then find the derivative of the function.

See Solution

Problem 3639

Bestimmen Sie die Wendepunkte von $f$ : a) $f(x)=x^{3}+2$ , b) $f(x)=4+2 x-x^{2}$ , e) $f(x)=\frac{1}{3} x^{3}-x^{2}+2 x$ .

See Solution

Problem 3640

Calculate the diver's speed when entering the water from an 82 ft platform, neglecting air resistance. Answer in mph.

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Problem 3641

Simplify the expression: $\frac{\left(\frac{1}{x+h}-\frac{1}{x}\right)}{h}$ .

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Problem 3642

Find the rate of change of vehicles in line for $f(x)=\frac{x^{2}}{3(2-x)}$ at $x=0.3$ . Round to four decimal places.

See Solution

Problem 3643

Simplify the expression: $\frac{\left(\frac{1}{x+h}-\frac{1}{x}\right)}{h}$ .

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Problem 3644

Berechnen Sie folgende Integrale: a) $\int_{0}^{2}(x^{3}-4 x) d x$ , b) $\int_{1}^{2}(x^{2}-3)^{2} d x$ , c) $\int_{-2}^{4} 3 d t$ , d) $\int_{0}^{\pi} \cos (x) d x$ , e) $\int_{0}^{\pi} \sin (t) d t$ , f) $\int_{1}^{9} \frac{1}{\sqrt{t}} d t$ , g) $\int_{1}^{2}(x^{3}-\frac{1}{x^{2}}) d x$ , h) $\int_{-2}^{2}(x+\sin (x)) d x$ .

See Solution

Problem 3645

Berechnen Sie die Integrale: a) $\int_{0}^{2}(x^{3}-4 x) d x$ und c) $\int_{-2}^{4} 3 \mathrm{~d} t$ .

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Problem 3646

Maximale Rechtecke: a) Finde die Seitenlängen eines Rechtecks mit Umfang 1 m, um die Fläche zu maximieren. b) Gilt das Ergebnis für alle Rechtecke?

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Problem 3647

Berechnen Sie die Fläche zwischen der Funktion $f$ und der $x$ -Achse für die Intervalle: a) $f(x)=x+3, I=[0; 4]$ , b) $f(x)=2x^{2}+1, I=[1; 2]$ , c) $f(x)=(2-x)^{2}, I=[1; 3]$ . Skizze anfertigen.

See Solution

Problem 3648

Evaluate the integral $\int_{0}^{1} \frac{x^{3}}{\sqrt{9-x^{2}}} d x$ .

See Solution

Problem 3649

Find the rate of change of vehicles in line for $f(x)=\frac{x^{2}}{3(2-x)}$ at $x=0.3$ . Round to four decimal places.

See Solution

Problem 3650

Find the derivative of $y=(7 x^{4}-2 x^{2}+3)^{4}$ . What is $u=g(x)$ if $y=f(u)$ ? $g(x)=$

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Problem 3651

Find the derivative of $f(x) = e^{-0.2x}(10x - x^2)$ .

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Problem 3652

Integrate $\tan^{-1}\left(x^{\frac{1}{2}}\right)$ with respect to $x$ .

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Problem 3653

Find $D_{x}(f[g(x)])$ at $x=3$ and $D_{x}(g[f(x)])$ at $x=1$ using the given function values and derivatives.

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Problem 3654

Find the tangent line equation for $f(x)=\sqrt{x^{2}+33}$ at $x=4$ . Calculate $f^{\prime}(x)$ .

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Problem 3655

Find the tangent line equation for $f(x)=\sqrt{x^{2}+55}$ at $x=3$ . $y=$ (Use $\mathrm{x}$ as the variable.)

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Problem 3656

Find the tangent line equation for $f(x)=x(x^{2}-4x+5)^{9}$ at $x=2$ . $y=$ (Use $\mathrm{x}$ as the variable.)

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Problem 3657

Differentiate the function $f=e^{u^{2}}$ .

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Problem 3658

Find the production level $x$ (up to 110 units) that minimizes the average cost given by $C(x)=0.2 x^{3}-24 x^{2}+1503 x+30,968$ . Round to one decimal place.

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Problem 3659

If the whitetail deer population grows at $11.5\%$ , how long to double it? Round to the nearest tenth.

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Problem 3660

Find the tangent line to $g(x)=\frac{16}{x}-4 \sqrt{x}$ at the point where $x=4$ .

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Problem 3661

Find the Taylor series for $h(x)=x^{3} \ln (1-3 x)$ . Discuss the critical point at $x=0$ and classify it.

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Problem 3662

Find $\lim_{x \rightarrow 3} h(x)$ if $f(x) \leq h(x) \leq g(x)$ for $1 \leq x \leq 5$ with $f(x)=\frac{x^{2}-25}{x^{2}-29}$ and $g(x)=\frac{x+5}{2x+4}$ .

See Solution

Problem 3663

A drone descends at $25.0 \mathrm{~m/s}$ . You drop a tennis ball from a 180 m building. What’s the ball's speed when it meets the drone?

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Problem 3664

Find the limits: $f(x)=\frac{x-2}{2|x-2|}$ as $x \to 2^-$ , $x \to 2^+$ , and $x \to 2$ . What's the discontinuity type?

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Problem 3665

Determine if the function $g(x)=\left\{\begin{array}{ll}3 x^{2}-5 x+1 & \text { if } x \leq 0 \\ \frac{1}{x(x+1)} & \text { if } x>0\end{array}\right.$ is continuous at $x=0$ . Choose true statement: (A), (B), (C), or (D).

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Problem 3666

Evaluate $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=2x^2+3$ .

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Problem 3667

A farmer has 648 feet of fencing for a rectangular area divided into two equal parts. Find the dimensions for maximum area.

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Problem 3668

A farmer has 162 feet of fencing for a rectangular area with two equal regions. Find dimensions to maximize the area.

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Problem 3669

Given the equation $x^{-8} \cdot z^{3}=27$ , find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(3)$ . Round to two digits.

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Problem 3670

Find $z$ for $x=1$ from $x^{-7} \cdot z^4=256$ , then find $z'$ , $z''$ , and $x'$ at $z=4$ . Round to 2 digits.

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Problem 3671

Calculate the integral of $\ln \sqrt{x}$ with respect to $x$ : $\int \ln \sqrt{x} \, dx$ .

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Problem 3672

Integrate $\int \sin^{2} x \cos^{3} x \, dx$ .

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Problem 3673

Find the derivative of the function defined by $4 x^{3}+4 x y-6 y^{4}=233$ and express it in the given form.

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Problem 3674

Find the derivative of the function defined by $4 x^{2}+4 x y-8 y^{4}=112$ and express it in the given format. Round answers to two digits.

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Problem 3675

Given the equation $x^{-5} \cdot z^{4}=16$ , find:
(a) $z(1)=$ (b) $z'(1)=$ (c) $z''(1)=$ (d) $x'(2)=$

See Solution

Problem 3676

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(2)$ for the equation $x^{-5} z^4 = 16$ . Round to two digits.

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Problem 3677

Find the derivative $\frac{d}{d u} v$ from $u^{8}+v^{8}=T$ in the form $A\left(\frac{u}{v}\right)^{a}+B(u \cdot v)^{b}+C$ . What are the coefficients $A$ , $a$ , $B$ , $b$ , and $C$ ?

See Solution

Problem 3678

Find $z(1)$ , $z'(1)$ , and $z''(1)$ for $x^{-5} \cdot z^{4}=16$ . Round to two decimal places.

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Problem 3679

Find the derivative of $k$ with respect to $l$ from $k^{10} + l^{10} = W$ and express it as $\frac{d}{d k} l=A\left(\frac{k}{l}\right)^{a}+B(k \cdot l)^{b}+C$ . Determine $A$ , $a$ , $B$ , $b$ , and $C$ .

See Solution

Problem 3680

Find the derivative of $y$ from the equation $5 x^{3}+6 x y-5 y^{4}=232$ and determine the coefficient $a$ .

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Problem 3681

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(7)$ for the equation $x^{-9} \cdot z^{3}=343$ . Round to two digits.

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Problem 3682

Given $f(5)=1$ , $f^{\prime}(5)=4$ , $g(5)=-3$ , and $g^{\prime}(5)=7$ , find: (a) $(f g)^{\prime}(5)$ (b) $\left(\frac{f}{g}\right)^{\prime}(5)$ (c) $\left(\frac{g}{f}\right)^{\prime}(5)$

See Solution

Problem 3683

Find the derivative $\frac{d}{d s} r$ from $r^{10}+s^{10}=Q$ and express it as $\frac{d}{d s} r=A\left(\frac{s}{r}\right)^{a}+B(s \cdot r)^{b}+C$ . What are the coefficients $A$ , $a$ , $B$ , $b$ , and $C$ ?

See Solution

Problem 3684

Find the integral of $x^{3} e^{x^{2}}$ with respect to $x$ .

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Problem 3685

Find the rate of biomass increase $B'(4)$ for a guppy population of 828, growing at 50/week, with avg mass $1.2$ g.

See Solution

Problem 3686

Berechne das Integral: $\int_{0}^{1}\left(-\frac{3 x}{\sqrt{k^{3}-3 x^{2}}}\right) d x$

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Problem 3687

Find the rate of change of $A$ with respect to $C$ given $B(C)=\cos C$ and $\frac{dA}{dB}=B^{-3}$ .

See Solution

Problem 3688

Find the elasticity of the function $t(z) = 29 z^{4} \ln z$ and determine the coefficients in the expression for $E l_{z} t(z)$ .

See Solution

Problem 3689

Find the elasticity and slope of the function $g(y) = 31 \cdot y^{-0.5}$ at $y_0 = 1$ . Round to two digits.

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Problem 3690

Find the derivative of $f(x)=3 x^{5}+5 x^{4}+6 x^{1/3}+5 x^{-3}+3 x+22$ and the coefficients $a$ to $k$ . What is $a$ ?

See Solution

Problem 3691

Given the function $f(x)=3 x^{5}+5 x^{4}+6 x^{1/3}+5 x^{-3}+3 x+22$ , find its derivative and coefficients $a$ to $k$ .

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Problem 3692

A ball is thrown from a building where $h(t)=25-t^{2}$ . Find: (a) building height, (b) time to hit ground, (c) avg. velocity in 2 sec, (d) avg. velocity in last 2 sec.

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Problem 3693

Find the derivative of $f(x)=3 x^{5}+5 x^{4}+6 x^{1/3}+5 x^{-3}+3 x+22$ and express it as $\frac{d f^{4}}{d x}(x)=a \cdot x^{6}+b \cdot x^{5}+c \cdot x^{4}+d \cdot x^{3}+e \cdot x^{1/3}+g \cdot x^{-2/3}+h \cdot x^{-3}+i \cdot x^{-4}+j \cdot x+k$ . Determine coefficients $a$ to $k$ .

See Solution

Problem 3694

Find the derivative of $g(f(x))$ at $x=7$ using $f(x)=3(x-6)^{2}-1$ and $g(y)=4y^{2}+3y$ .

See Solution

Problem 3695

Find the integral of $\frac{x}{x^{2}+3}$ with respect to $x$ .

See Solution

Problem 3696

Evaluate $\lim _{x \rightarrow 1} g(x)$ given $2 x \leq g(x) \leq x^{4}-x^{2}+2$ for all $x$ .

See Solution

Problem 3697

A ball is thrown from a building with height $h(t)=25-t^{2}$ . Find the building height, time to hit ground, and average velocities.

See Solution

Problem 3698

Find $\lim _{x \rightarrow 0} f(x)$ if $\lim _{x \rightarrow 0}(2 f(x)-15)=45$ .

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Problem 3699

Determine if the following statements are true or false, and provide explanations or examples:
a) If $\lim _{x \rightarrow 0} f(x)=0$ and $\lim _{x \rightarrow 0} g(x)=0$ , then $\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}$ does not exist.
b) If $\lim _{x \rightarrow a} f(x)=L$ , then $f(a)=L$ .

See Solution

Problem 3700

Find the rate of change of $A$ with respect to $C$ given $B(C)=\cos C$ and $\frac{dA}{dB}=B^{-3}$ .

See Solution

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Calculus

Problem 3601

Find the surface area of the graph y=2x2y=2 x^{2}y=2x2 from x=2x=2x=2 to x=4x=4x=4 when revolved around the xxx-axis. Set up the integral.

Problem 3602

A particle moves along the curve r⃗(t)=(cos⁡∧2t,−cos⁡∧2t,t)\vec{r}(t)=\left(\cos ^{\wedge} 2 t,-\cos ^{\wedge} 2 t, t\right)r(t)=(cos∧2t,−cos∧2t,t). Find its max speed and when it occurs, plus the acceleration related to direction change.

Problem 3603

Stilus-Stifte hat eine Kostenfunktion K(x)=2x3−18x2+62x+32K(x)=2 x^{3}-18 x^{2}+62 x+32K(x)=2x3−18x2+62x+32. a) Erklären Sie den Graphen und den Wendepunkt. b) Bei \$50 pro 1000 Bleistifte, wo wird Gewinn erzielt und wo ist er maximal?

Problem 3604

Find the infinite limits: 1. lim⁡x→3+8x−3\lim _{x \rightarrow 3^{+}} \frac{8}{x-3}limx→3+​x−38​; I. lim⁡x→3−8x−3\lim _{x \rightarrow 3^{-}} \frac{8}{x-3}limx→3−​x−38​.

Problem 3605

Calculate the integral: ∫x2−1x2+1dx\int \frac{x^{2}-1}{x^{2}+1} d x∫x2+1x2−1​dx

Problem 3606

Evaluate the integral: ∫2x2+1 dx\int \frac{2}{x^{2}+1} \, dx∫x2+12​dx.

Problem 3607

Find the direction of r′r^{\prime}r′ for r(t)=⟨5t−2,7t+1,t2⟩r(t)=\langle 5t-2, 7t+1, t^{2} \rangler(t)=⟨5t−2,7t+1,t2⟩ at t0=1t_{0}=1t0​=1. Choose the correct option.

Problem 3608

Evaluate the limit as hhh approaches 0: lim⁡h→02(3+h)2−18h\lim_{h \rightarrow 0} \frac{2(3+h)^{2}-18}{h}limh→0​h2(3+h)2−18​. Simplify first.

Problem 3609

Problem 3610

Problem 3611

Calculate the integral: ∫x2+1x2−1dx\int \frac{x^{2}+1}{x^{2}-1} d x∫x2−1x2+1​dx

Problem 3612

Problem 3613

Calculate the integral: ∫1x2+8x+7dx\int \frac{1}{\sqrt{x^{2}+8 x+7}} d x∫x2+8x+7​1​dx

Problem 3614

Evaluate the integral ∫1v=9−4x2dx\int \frac{1}{v=9-4 x^{2}} d x∫v=9−4x21​dx.

Problem 3615

Find the difference quotient, f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}hf(a+h)−f(a)​, for f(x)=x2−xf(x)=\sqrt{x^{2}-x}f(x)=x2−x​.

Problem 3616

Evaluate the integral: ∫149−4x2dx\int \frac{1}{\sqrt{49-4 x^{2}}} d x∫49−4x2​1​dx

Problem 3617

Find the integral of 1x(x+2)\frac{1}{x(x+2)}x(x+2)1​ with respect to xxx.

Problem 3618

Calculate the integral ∫x3x2+1dx\int \frac{x^{3}}{x^{2}+1} d x∫x2+1x3​dx.

Problem 3619

Bestimmen Sie die Ableitung von f(x)=x3+xf(x)=x^{3}+\sqrt{x}f(x)=x3+x​.

Problem 3620

Bestimmen Sie die Extrempunkte der Funktion f(x)=12x3−axf(x) = \frac{1}{2} x^{3}-a xf(x)=21​x3−ax.

Problem 3621

Find the derivative using the product rule for: p(y)=(y−1+y−2)(2y−3−7y−4)p(y)=\left(y^{-1}+y^{-2}\right)\left(2 y^{-3}-7 y^{-4}\right)p(y)=(y−1+y−2)(2y−3−7y−4). What is p′(y)p^{\prime}(y)p′(y)?

Problem 3622

Problem 3623

Calculate the limit: lim⁡x→∞ln⁡xx\lim _{x \rightarrow \infty} \frac{\ln x}{\sqrt{x}}limx→∞​x​lnx​.

Problem 3624

Find h′(8)h^{\prime}(8)h′(8) for h(x)=f(x)g(x)h(x)=\frac{f(x)}{g(x)}h(x)=g(x)f(x)​ given f(8)=1f(8)=1f(8)=1, f′(8)=2f'(8)=2f′(8)=2, g(8)=9g(8)=9g(8)=9, g′(8)=6g'(8)=6g′(8)=6.

Problem 3625

Find the tangent line equation to f(x)=(3x−2)(x+6)f(x)=(3 x-2)(x+6)f(x)=(3x−2)(x+6) at the point (1,7)(1,7)(1,7).

Problem 3626

Find the tangent line equation for f(x)=(3x−5)(x+6)f(x)=(3x-5)(x+6)f(x)=(3x−5)(x+6) at the point (2,8)(2,8)(2,8).

Problem 3627

Find the derivative of the function f(x)=x3+84f(x)=\frac{x^{3}+8}{4}f(x)=4x3+8​.

Problem 3628

5 Forscher untersuchen das Bakterienwachstum mit A(t)=−0,005t3+0,2t2+0,9t+1A(t)=-0,005 t^{3}+0,2 t^{2}+0,9 t+1A(t)=−0,005t3+0,2t2+0,9t+1. a) Fläche um 3 Uhr morgens? b) Maximale Zunahme der Fläche?

Problem 3629

Find the values of xxx where f′(x)=0f'(x)=0f′(x)=0 for the function f(x)=(x2−5)(x2−5)f(x)=(x^{2}-5)(x^{2}-\sqrt{5})f(x)=(x2−5)(x2−5​).

Problem 3630

Calculate the derivative of the product f(x)g(x)f(x) g(x)f(x)g(x) using f′(x)g′(x)f'(x) g'(x)f′(x)g′(x) for f(x)=x2+bxf(x)=x^{2}+b xf(x)=x2+bx and g(x)=cx+dg(x)=c x+dg(x)=cx+d. What is f′(x)g′(x)f'(x) g'(x)f′(x)g′(x)?

Problem 3631

Calculate the derivatives: (a) Find f′(x)g′(x)f^{\prime}(x) g^{\prime}(x)f′(x)g′(x) for f(x)=x2+bxf(x)=x^{2}+bxf(x)=x2+bx and g(x)=cx+dg(x)=cx+dg(x)=cx+d. (b) Compute [f(x)g(x)]′[f(x) g(x)]^{\prime}[f(x)g(x)]′ and show it differs from part (a).

Problem 3632

Find the derivative of the function f(x)=x9f(x)=x^{9}f(x)=x9.

Problem 3633

Bestimmen Sie die Ableitung von g(x)=x2ng(x)=x^{2 n}g(x)=x2n.

Problem 3634

A bike company finds that a new employee assembles M(d)=97d24d2+11M(d)=\frac{97 d^{2}}{4 d^{2}+11}M(d)=4d2+1197d2​ bikes/day after ddd days. (a) Determine M′(d)M'(d)M′(d). (b) Calculate and explain M′(2)M'(2)M′(2) and M′(5)M'(5)M′(5).

Problem 3635

Find the derivative of f(x)=x7f(x)=x^{7}f(x)=x7 at the point x0=−1x_{0}=-1x0​=−1.

Problem 3636

Find the derivative of the function g(x)=x−2g(x)=x^{-2}g(x)=x−2 using the power rule.

Problem 3637

Find the derivative s′(x)s'(x)s′(x) of the function s(x)=xm+nxs(x) = \frac{x}{m+nx}s(x)=m+nxx​ and evaluate it at x=50x=50x=50, m=15m=15m=15, n=3n=3n=3.

Problem 3638

Rewrite f(x)=1x7f(x)=\frac{1}{x^{7}}f(x)=x71​ with xxx in the numerator, then find the derivative of the function.

Problem 3639

Bestimmen Sie die Wendepunkte von fff: a) f(x)=x3+2f(x)=x^{3}+2f(x)=x3+2, b) f(x)=4+2x−x2f(x)=4+2 x-x^{2}f(x)=4+2x−x2, e) f(x)=13x3−x2+2xf(x)=\frac{1}{3} x^{3}-x^{2}+2 xf(x)=31​x3−x2+2x.

Problem 3640

Calculate the diver's speed when entering the water from an 82 ft platform, neglecting air resistance. Answer in mph.

Problem 3641

Simplify the expression: (1x+h−1x)h\frac{\left(\frac{1}{x+h}-\frac{1}{x}\right)}{h}h(x+h1​−x1​)​.

Problem 3642

Find the surface area of the graph $y=2 x^{2}$ from $x=2$ to $x=4$ when revolved around the $x$ -axis. Set up the integral.

A particle moves along the curve $\vec{r}(t)=\left(\cos ^{\wedge} 2 t,-\cos ^{\wedge} 2 t, t\right)$ . Find its max speed and when it occurs, plus the acceleration related to direction change.

Stilus-Stifte hat eine Kostenfunktion $K(x)=2 x^{3}-18 x^{2}+62 x+32$ . a) Erklären Sie den Graphen und den Wendepunkt. b) Bei \$50 pro 1000 Bleistifte, wo wird Gewinn erzielt und wo ist er maximal?

Find the infinite limits: 1. $\lim _{x \rightarrow 3^{+}} \frac{8}{x-3}$ ; I. $\lim _{x \rightarrow 3^{-}} \frac{8}{x-3}$ .

Calculate the integral: $\int \frac{x^{2}-1}{x^{2}+1} d x$

Evaluate the integral: $\int \frac{2}{x^{2}+1} \, dx$ .

Find the direction of $r^{\prime}$ for $r(t)=\langle 5t-2, 7t+1, t^{2} \rangle$ at $t_{0}=1$ . Choose the correct option.

Evaluate the limit as $h$ approaches 0: $\lim_{h \rightarrow 0} \frac{2(3+h)^{2}-18}{h}$ . Simplify first.

Calculate the integral: $\int \frac{x^{2}+1}{x^{2}-1} d x$

Calculate the integral: $\int \frac{1}{\sqrt{x^{2}+8 x+7}} d x$

Evaluate the integral $\int \frac{1}{v=9-4 x^{2}} d x$ .

Find the difference quotient, $\frac{f(a+h)-f(a)}{h}$ , for $f(x)=\sqrt{x^{2}-x}$ .

Evaluate the integral: $\int \frac{1}{\sqrt{49-4 x^{2}}} d x$

Find the integral of $\frac{1}{x(x+2)}$ with respect to $x$ .

Calculate the integral $\int \frac{x^{3}}{x^{2}+1} d x$ .

Bestimmen Sie die Ableitung von $f(x)=x^{3}+\sqrt{x}$ .

Bestimmen Sie die Extrempunkte der Funktion $f(x) = \frac{1}{2} x^{3}-a x$ .

Find the derivative using the product rule for: $p(y)=\left(y^{-1}+y^{-2}\right)\left(2 y^{-3}-7 y^{-4}\right)$ . What is $p^{\prime}(y)$ ?

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{\ln x}{\sqrt{x}}$ .

Find $h^{\prime}(8)$ for $h(x)=\frac{f(x)}{g(x)}$ given $f(8)=1$ , $f'(8)=2$ , $g(8)=9$ , $g'(8)=6$ .

Find the tangent line equation to $f(x)=(3 x-2)(x+6)$ at the point $(1,7)$ .

Find the tangent line equation for $f(x)=(3x-5)(x+6)$ at the point $(2,8)$ .

Find the derivative of the function $f(x)=\frac{x^{3}+8}{4}$ .

5 Forscher untersuchen das Bakterienwachstum mit $A(t)=-0,005 t^{3}+0,2 t^{2}+0,9 t+1$ .
a) Fläche um 3 Uhr morgens? b) Maximale Zunahme der Fläche?

Find the values of $x$ where $f'(x)=0$ for the function $f(x)=(x^{2}-5)(x^{2}-\sqrt{5})$ .

Calculate the derivative of the product $f(x) g(x)$ using $f'(x) g'(x)$ for $f(x)=x^{2}+b x$ and $g(x)=c x+d$ . What is $f'(x) g'(x)$ ?

Calculate the derivatives:
(a) Find $f^{\prime}(x) g^{\prime}(x)$ for $f(x)=x^{2}+bx$ and $g(x)=cx+d$ .
(b) Compute $[f(x) g(x)]^{\prime}$ and show it differs from part (a).

Find the derivative of the function $f(x)=x^{9}$ .

Bestimmen Sie die Ableitung von $g(x)=x^{2 n}$ .

A bike company finds that a new employee assembles $M(d)=\frac{97 d^{2}}{4 d^{2}+11}$ bikes/day after $d$ days.
(a) Determine $M'(d)$ . (b) Calculate and explain $M'(2)$ and $M'(5)$ .

Find the derivative of $f(x)=x^{7}$ at the point $x_{0}=-1$ .

Find the derivative of the function $g(x)=x^{-2}$ using the power rule.

Find the derivative $s'(x)$ of the function $s(x) = \frac{x}{m+nx}$ and evaluate it at $x=50$ , $m=15$ , $n=3$ .

Rewrite $f(x)=\frac{1}{x^{7}}$ with $x$ in the numerator, then find the derivative of the function.

Bestimmen Sie die Wendepunkte von $f$ : a) $f(x)=x^{3}+2$ , b) $f(x)=4+2 x-x^{2}$ , e) $f(x)=\frac{1}{3} x^{3}-x^{2}+2 x$ .

Simplify the expression: $\frac{\left(\frac{1}{x+h}-\frac{1}{x}\right)}{h}$ .

Find the rate of change of vehicles in line for $f(x)=\frac{x^{2}}{3(2-x)}$ at $x=0.3$ . Round to four decimal places.

Simplify the expression: $\frac{\left(\frac{1}{x+h}-\frac{1}{x}\right)}{h}$ .

Berechnen Sie die Integrale: a) $\int_{0}^{2}(x^{3}-4 x) d x$ und c) $\int_{-2}^{4} 3 \mathrm{~d} t$ .

Berechnen Sie die Fläche zwischen der Funktion $f$ und der $x$ -Achse für die Intervalle: a) $f(x)=x+3, I=[0; 4]$ , b) $f(x)=2x^{2}+1, I=[1; 2]$ , c) $f(x)=(2-x)^{2}, I=[1; 3]$ . Skizze anfertigen.

Evaluate the integral $\int_{0}^{1} \frac{x^{3}}{\sqrt{9-x^{2}}} d x$ .

Find the rate of change of vehicles in line for $f(x)=\frac{x^{2}}{3(2-x)}$ at $x=0.3$ . Round to four decimal places.

Find the derivative of $y=(7 x^{4}-2 x^{2}+3)^{4}$ . What is $u=g(x)$ if $y=f(u)$ ? $g(x)=$

Find the derivative of $f(x) = e^{-0.2x}(10x - x^2)$ .

Integrate $\tan^{-1}\left(x^{\frac{1}{2}}\right)$ with respect to $x$ .

Find $D_{x}(f[g(x)])$ at $x=3$ and $D_{x}(g[f(x)])$ at $x=1$ using the given function values and derivatives.

Find the tangent line equation for $f(x)=\sqrt{x^{2}+33}$ at $x=4$ . Calculate $f^{\prime}(x)$ .

Find the tangent line equation for $f(x)=\sqrt{x^{2}+55}$ at $x=3$ . $y=$ (Use $\mathrm{x}$ as the variable.)

Find the tangent line equation for $f(x)=x(x^{2}-4x+5)^{9}$ at $x=2$ . $y=$ (Use $\mathrm{x}$ as the variable.)

Differentiate the function $f=e^{u^{2}}$ .

Find the production level $x$ (up to 110 units) that minimizes the average cost given by $C(x)=0.2 x^{3}-24 x^{2}+1503 x+30,968$ . Round to one decimal place.

If the whitetail deer population grows at $11.5\%$ , how long to double it? Round to the nearest tenth.

Find the tangent line to $g(x)=\frac{16}{x}-4 \sqrt{x}$ at the point where $x=4$ .

Find the Taylor series for $h(x)=x^{3} \ln (1-3 x)$ . Discuss the critical point at $x=0$ and classify it.

Find $\lim_{x \rightarrow 3} h(x)$ if $f(x) \leq h(x) \leq g(x)$ for $1 \leq x \leq 5$ with $f(x)=\frac{x^{2}-25}{x^{2}-29}$ and $g(x)=\frac{x+5}{2x+4}$ .

A drone descends at $25.0 \mathrm{~m/s}$ . You drop a tennis ball from a 180 m building. What’s the ball's speed when it meets the drone?

Find the limits: $f(x)=\frac{x-2}{2|x-2|}$ as $x \to 2^-$ , $x \to 2^+$ , and $x \to 2$ . What's the discontinuity type?

Evaluate $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=2x^2+3$ .

Given the equation $x^{-8} \cdot z^{3}=27$ , find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(3)$ . Round to two digits.

Find $z$ for $x=1$ from $x^{-7} \cdot z^4=256$ , then find $z'$ , $z''$ , and $x'$ at $z=4$ . Round to 2 digits.

Calculate the integral of $\ln \sqrt{x}$ with respect to $x$ : $\int \ln \sqrt{x} \, dx$ .

Integrate $\int \sin^{2} x \cos^{3} x \, dx$ .

Find the derivative of the function defined by $4 x^{3}+4 x y-6 y^{4}=233$ and express it in the given form.

Find the derivative of the function defined by $4 x^{2}+4 x y-8 y^{4}=112$ and express it in the given format. Round answers to two digits.