Calculus

Problem 30401

Verkaufsprognose: Analysiere die Funktion $f(t)=t^{3}-24 t^{2}+150 t+100$ für Downloads in Tausend. Beantworte Fragen a) bis g).

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Problem 30402

Find the rate of change of average cost when producing 179 belts, given $C(x)=710+32x-0.062x^{2}$ .

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Problem 30403

Find the rate of change of average cost when producing 179 belts, given $C(x)=710+32x-0.062x^{2}$ . Calculate $\bar{C}^{\prime}(x)$ .

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Problem 30404

At an excavation site, the dirt removed is modeled by $f(t)$ where $g(t)$ is given.
(a) Find the average rate of change of $f$ from $t=6$ to $t=12$ . (b) Approximate $f^{\prime}(9)$ using the table data. Show your calculations.

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Problem 30405

Is the function $f$ continuous for $0 \leq t \leq 12$ ? Justify. Also, find $f^{\prime}(2)$ for dirt removed at $t=2$ hours.

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Problem 30406

The temperature $T(t)=\frac{8 t}{t^{2}+4}+98.6$ gives (a) the rate of change, (b) $T(1)$ , (c) change rate at $t=1$ .

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Problem 30407

Find the average cost change rate when producing 179 belts, given $C(x)=750+35x-0.068x^{2}$ .

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Problem 30408

Find the rate of change of average cost for producing 179 belts with $C(x)=750+35x-0.068x^{2}$ .

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Problem 30409

Find the temperature function $T(t)=\frac{8 t}{t^{2}+4}+98.6$ . (a) Determine $\frac{dT}{dt}$ . (b) Calculate $T(1)$ . (c) Find $\frac{dT}{dt}$ at $t=1$ .

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Problem 30410

Find the temperature function $T(t)=\frac{8t}{t^{2}+4}+98.6$ . (a) Determine $T'(t)$ . (b) Calculate $T(1)$ . (c) Find $T'(1)$ .

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Problem 30411

Find the derivative of $y=\frac{x^{9}}{x^{3}}$ using the Quotient Rule and select the correct answer.

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Problem 30412

Given the temperature function $T(t)=\frac{8 t}{t^{2}+4}+98.6$ , find: (a) $T'(t)$ , (b) $T(1)$ , (c) $T'(1)$ .

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Problem 30413

Find the derivative of $y=\frac{x^{9}}{x^{3}}$ using the Quotient Rule and by dividing first.

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Problem 30414

Find the derivative of $y=\frac{x^{9}}{x^{3}}$ using the Quotient Rule and select the correct answer.

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Problem 30415

Divide $x^{9}$ by $x^{3}$ , simplify to $x^{6}$ , then find and simplify $\frac{d y}{d x}=\square$ .

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Problem 30416

Is the sequence $a_n$ with $|a_n - a_{n+1}| \leq \frac{1}{n}$ always convergent? Also, find a formula for $\sum_{k=1}^{n} \frac{1}{(4k+3)(4k+7)}$ and compute $\sum_{k=1}^{\infty} \frac{1}{(4k+3)(4k+7)}$ .

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Problem 30417

Find the derivative of $y=\frac{x^{9}}{x^{3}}$ using the Quotient Rule. Choose the correct derivative form from A, B, C, or D.

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Problem 30418

Find the derivative of $y=\frac{x^{6}}{x^{4}}$ using the Quotient Rule and complete the answer choices.

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Problem 30419

Cobalt-60 decays over time. Use the model $A(t)=A_{0}\left(\frac{1}{2}\right)^{\frac{t}{5.3}}$ to answer:
a) How long to reduce to half? b) Percent remaining after 10 years? c) Time until $10\%$ remains?

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Problem 30420

Determinați dimensiunile unei săli de cinema cu volum de 4000 m³ pentru a minimiza $f(x, y, z) = xy + 2yz + 2xz$ cu $xyz = 4000$ . Folosiți metoda multiplicatorilor Lagrange.

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Problem 30421

Find the derivative of $y=\frac{x^{6}}{x^{4}}$ using the Quotient Rule and select the correct answer.

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Problem 30422

Estimate the rate of change of $f(x)=\frac{x}{x-4}$ at the point $(2,-1)$ .

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Problem 30423

Find a formula for the sum $\sum_{k=1}^{n} \frac{1}{(4 k+3)(4 k+7)}$ and compute $\sum_{k=1}^{\infty} \frac{1}{(4 k+3)(4 k+7)}$ .

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Problem 30424

Divide $x^{8}$ by $x^{3}$ , simplify to get $x^{5}$ . Then find and simplify the derivative $\frac{d y}{d x}$ .

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Problem 30425

Find the acceleration of a particle with velocity $v(t)=3+4.1 \cos (0.9 t)$ at time $t=4$ . Choices: (A) -2.016 (B) -0.677 (C) 1.633 (D) 1.814 (E) 2.978

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Problem 30426

Divide $x^{6}$ by $x^{4}$ , simplify to $x^{2}$ . Find its derivative and verify $\frac{d y}{d x}=\square$ .

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Problem 30427

Calculați următoarele integrale folosind proprietățile integralelor euleriene: 1) $\int_{0}^{1} \sqrt{x}(x-1)^{2} d x$ ; 2) $\int_{0}^{1} \frac{x-1}{\sqrt{x}} d x$ ; 3) $\int_{0}^{1} \frac{(x-1)^{3}}{\sqrt[4]{x}} d x$ ; 4) $\int_{0}^{1} \frac{1-x^{2}}{\sqrt{x}} d x$ ; 5) $\int_{0}^{1} \frac{1+2 x+3 x^{2}}{\sqrt{1-x}} d x$ ; 6) $\int_{0}^{1}(x+1)^{2} \sqrt[3]{1-x} d x$ ; 7) $\int_{0}^{1} \frac{d x}{\sqrt[99]{x-1}}$ .

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Problem 30428

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{2}-\cos x-2 x}{x^{2}-2 x}$ . Options: (A) $-\frac{1}{2}$ (B) 0 (C) $\frac{1}{2}$ (D) 1 (E) nonexistent.

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Problem 30429

Find the slope of the tangent line to $x^{2}+x y+y^{2}=7$ at the point $(2,1)$ . Choices: (A) $-\frac{4}{3}$ (B) $-\frac{5}{4}$ (C) -1 (D) $-\frac{4}{5}$ (E) $-\frac{3}{4}$ .

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Problem 30430

Find the values of $t$ where the particle's position $x(t)=t^{3}-3 t^{2}-9 t+1$ is at rest.

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Problem 30431

Divide $x^{5}/x^{3}$ and find the derivative of the result. Verify that $\frac{d y}{d x}=\square$ .

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Problem 30432

Calculați integralele: 3) $\int_{0}^{1} \frac{(x-1)^{3}}{\sqrt[4]{x}} d x$ ; 4) $\int_{0}^{1} \frac{1-x^{2}}{\sqrt{x}} d x$ .

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Problem 30433

Differentiate $y=12 \sqrt{x}$ . Find $\frac{d}{d x}(12 \sqrt{x})=\square$ .

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Problem 30434

Find the derivative $\frac{d y}{d x}$ for the function $y=-\frac{10}{x^{8}}$ . What is $\frac{d y}{d x}=?$

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Problem 30435

Differentiate the function: $y=4 x^{3}-5 x^{2}+8 x+9$ . Find $\frac{d y}{d x}$ .

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Problem 30436

Find a formula for $\sum_{k=1}^{n}(k+3) q^{k}$ with $|q|<1$ , and compute $\sum_{k=1}^{\infty}(k+3) q^{k}$ .

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Problem 30437

Find the derivative of the function: $y=7 x^{-\frac{5}{2}}+8 x^{-\frac{1}{2}}+x^{5}-8$ . What is $y'$ ?

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Problem 30438

Differentiate the function $y=\frac{x}{11}-\frac{11}{x}$ . Find $\frac{d}{dx}\left(\frac{x}{11}-\frac{11}{x}\right)$ .

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Problem 30439

Find where the tangent line to $y=-0.025 x^{2}+5 x$ has a slope of 9. The point(s) is/are $\square$ .

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Problem 30440

Differentiate the function $y=\frac{4 x^{2}-3}{2 x^{3}+7}$ . Find $y^{\prime}$ .

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Problem 30441

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2 x^{2}-4 x+1$ and calculate $f^{\prime}(-1)$ .

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Problem 30442

Find the derivative of $f(x)=(x-7)(2x+5)$ using the Product Rule and by expanding the product.

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Problem 30443

The function $s(t)=8 t(t+2)$ gives distance in km after time $t$ in hours (0 ≤ $t$ ≤ 5). Find average velocity for: a. i. $t=3$ to $t=4$ ; ii. $t=3$ to $t=3.1$ ; iii. $t=3$ to $t=3.01$ . b. Approximate instantaneous velocity at $t=3$ . c. Calculate velocity at $t=3$ .

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Problem 30444

Given the equation $R(v)=\frac{6400}{v}$ , find: a) $R'(v)$ , b) $R(90)$ , c) $R'(90)$ . Round as needed.

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Problem 30445

Differentiate the function $F(x)=(x^{5}-8 x)^{2}$ . Find $F^{\prime}(x)=\square$ .

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Problem 30446

Find the rate of change of average revenue when 400 jackets are produced, given $R(x)=85 \sqrt{x}$ .

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Problem 30447

Differentiate the function $y=\sqrt{x^{2}+11}$ . Find $\frac{d y}{d x}=\square$ .

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Problem 30448

Find the derivative of the function $f(x) = 2 e^{x} - \frac{1}{4} x^{4}$ . What is $f^{\prime}(x)$ ?

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Problem 30449

Differentiate the function $f(x)=\sqrt{x}-(x-1)^{3}$ . Find $f^{\prime}(x)=$ .

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Problem 30450

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}+x+7}-3}{x^{2}-6 x+5}$ .

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Problem 30451

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

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Problem 30452

Berechne $U_{n}$ und $U_{\underline{O}}$ für $f(x)=x^{2}$ im Intervall $[0 ; 1]$ , unterteilt in $n$ Teilintervalle.

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Problem 30453

Differentiate the function $f(x)=\left(\frac{x-4}{x+3}\right)^{6}$ . Find $f^{\prime}(x)=$ .

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Problem 30454

A student learns $N(t)=20t-t^{2}$ terms after $t$ hours. Find terms learned from $t=2$ to $t=3$ and rate at $t=2$ .

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Problem 30455

Find the first and second derivatives of the function $f(x) = 3 e^{x} + 2 x^{3} - x + 5$ .

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Problem 30456

Differentiate the function $f(x)=(2 x^{5}-5 x^{3}+3)^{26}$ . Find $f^{\prime}(x)=$ .

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Problem 30457

Find the first and second derivatives of the function $f(x) = \frac{2}{x} - 3 e^{x}$ .

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Problem 30458

Find the first and second derivatives of the function $f(x) = 0.3 e^{x} - \sqrt{x}$ .

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Problem 30459

Find the rate of change of the particle's position given $s=\sqrt{45+6t}$ at $t=6 \mathrm{sec}$ . Answer: $\square \mathrm{m/sec}$ .

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Problem 30460

Bestimme die Ableitungen für folgende Funktionen: a) $f(x)=e^{x}-a x^{2}$ b) $f(x)=a e^{x}+x^{b}$ c) $f(x)=\frac{e^{x}}{a}+b^{2} x^{2}$ d) $f(x)=-a e^{x}+a$ $f^{\prime}(x)=$ $f^{\prime}(x)=$ $f^{\prime}(x)=$ $f^{\prime}(x)=$

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Problem 30461

Find $y^{\prime}$ if $y=\frac{x \ln (x)}{1+\ln (x)}$ . Choose from options a, b, c, d, or e.

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Problem 30462

Find the derivative $\frac{d C}{d x}$ of the function $C(x)=9.16 x^{4}-85.05 x^{3}+287.45 x^{2}-309.42 x+2651.5$ . Interpret its meaning.

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Problem 30463

Gegeben ist die Funktion $f_k(x) = x^2 + 2kx + 2x + 1$ . Finde die Tiefpunkte und deren Koordinaten.

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Problem 30464

Find the second derivative $f^{\prime \prime}(x)$ for the function $f(x)=3 x^{2}-11 x-\frac{5}{x^{2}}$ .

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Problem 30465

Model consumer credit with $C(x)=9.16 x^{4}-85.05 x^{3}+287.45 x^{2}-309.42 x+2651.5$ . Estimate rise in 2015.

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Problem 30466

Given $s(t)=5 t^{2}+20 t$ , find: a) $v(t)$ , b) $a(t)$ , c) velocity and acceleration at $t=2$ sec.

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Problem 30467

A medicine's amount $M(t)=-\frac{1}{3} t^{2}+t$ in blood changes over time. Find $M'(2)$ and explain if it's negative.

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Problem 30468

An object falls from rest; find (a) distance $s(5)$ , (b) speed $v(5)$ , and (c) acceleration after $5$ seconds.

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Problem 30469

Find the rate of change of temperature $T(h)=\frac{60}{h+2}$ with respect to height at $h=3$ km.

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Problem 30470

Ein Tank hat $10 \mathrm{~m}^{3}$ Flüssigkeit.
(a) Zuflussrate: $2 \frac{\mathrm{m}^{3}}{\mathrm{min}}$ bis $6 \frac{\mathrm{m}^{3}}{\mathrm{min}}$ (2. Minute), dann $0 \frac{\mathrm{m}^{3}}{\mathrm{min}}$ (8. Minute). Wie viel Flüssigkeit ist am Ende im Tank?
(b) Abflussrate: $3 \frac{\mathrm{m}^{3}}{\mathrm{min}}$ . Wie lange dauert es, bis der Tank leer ist?

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Problem 30471

Find the second derivative $f_{k}^{\prime \prime}(-k-1)$ for $f_{k}(x) = x^2 + 2kx + 2x + 1$ .

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Problem 30472

Find the derivatives of the following functions:
1. a) $f(x)=2x+\frac{1}{3}x^{6}$ b) $f(x)=x^{\frac{3}{2}}-5$ c) $f(x)=x+4x^{\frac{1}{2}}$ d) $f(x)=6x^{2}-x^{-4}$ e) $f(x)=7+x^{-\frac{4}{5}}$ f) $f(x)=2x^{\frac{1}{6}}+x^{\frac{1}{4}}$ g) $f(x)=3x^{-1}-5x^{-\frac{3}{2}}$ h) $f(x)=2-7x^{-1}+x^{-4}$
2. Find $\frac{dy}{dx}$ for: a) $y=\sqrt{x}$ b) $y=4-\frac{1}{x}$ c) $y=3x^{2}+\sqrt[3]{x}$ d) $y=9x+\frac{3}{x}$ e) $y=\frac{1}{4x}-\frac{1}{x^{2}}$ f) $y=\frac{6}{\sqrt[4]{x}}$ g) $y=\sqrt{x^{5}}$ h) $y=8\sqrt{x}+\frac{4}{3x^{2}}$

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Problem 30473

Find the derivative $\frac{\mathrm{d} y}{\mathrm{dx}}$ for each function: a) $y=\sqrt{x}$ b) $y=4-\frac{1}{x}$ c) $y=3 x^{2}+\sqrt[3]{x}$ d) $y=9 x+\frac{3}{x}$ e) $y=\frac{1}{4 x}-\frac{1}{x^{2}}$ f) $y=\frac{6}{\sqrt[4]{x}}$ g) $y=\sqrt{x^{5}}$ h) $y=8 \sqrt{x}+\frac{4}{3 x^{2}}$

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Problem 30474

Calculate the Riemann sum $R_{n}$ for $f(x)=-\frac{x^{2}}{3}-2$ on $[0,3]$ as a function of $n$ . Find $\lim_{n \to \infty} R_{n}$ .

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Problem 30475

Find $g^{\prime}(3)$ for $g(x)=\int_{x^{2}}^{7}(5 t^{3}-4 t+3) dt$ .

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Problem 30476

Approximate the integral $\int_{-2}^{5}(8 x-4) d x$ using the Midpoint Rule with $n$ subintervals. Find the limit as $n \to \infty$ .

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Problem 30477

Find $g^{\prime}(x)$ for $g(x)=\int_{x^{2}}^{8} \frac{e^{4 x}}{\sin x} d x$ . Choices: $-\frac{x e^{4 x^{2}}}{\sin x^{2}}$ , $-\frac{e^{4 x}}{\sin x}$ , $-\frac{e^{4 x^{2}}}{\sin x^{2}}$ , $-\frac{2 x e^{4 x^{2}}}{\sin x^{2}}$ .

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Problem 30478

Find $h^{\prime}(x)$ if $h(x)=\int_{1}^{\sqrt{x}} \frac{5 z^{2}}{z^{4}+1} d z$ .

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Problem 30479

Evaluate the integral from 3 to 5: $\int_{3}^{5} 5 \, dx = \square$

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Problem 30480

1.) Find the particle's acceleration given its velocities at $t=3.5 \mathrm{~s}$ and $t=10.0 \mathrm{~s}$ .
2.) A rock is thrown up at $40 \mathrm{~m/s}$ and lands on a platform $2.5 \mathrm{~m}$ high. a. What is the landing velocity? b. How long is it in the air? c. What is the max height above ground?
3.) Tom walks $20 \mathrm{~m}$ south and $70 \mathrm{~m}$ east. What is his displacement in unit vector notation?

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Problem 30481

Find the derivative $f^{\prime}(x)$ for $f(x)=5 \sqrt{x}$ and evaluate it at $x=6$ .

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Problem 30482

Find critical points for each function and classify them as local max, min, or horizontal tangent: a. $h(x)=-6 x^{3}+18 x^{2}+3$ b. $g(t)=t^{5}+t^{3}$ c. $y=(x-5)^{\frac{1}{3}}$ d. $f(x)=\left(x^{2}-1\right)^{\frac{1}{3}}$

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Problem 30483

Find the derivative $f^{\prime}(49)$ for $f(x)=\sqrt{x}$ using $f^{\prime}(c)=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$ .

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Problem 30484

Find the derivative $f^{\prime}(x)$ for $f(x)=\frac{-11}{x}$ at $x=-4$ .

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Problem 30485

Find $\frac{d t}{d x}$ for $t=\frac{x}{7 x-9}$ .

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Problem 30486

Find critical points of $f(x)=\left(x^{2}-1\right)^{\frac{1}{3}}$ and classify them as local max, min, or horizontal tangent.

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Problem 30487

Find the critical points of $f(x)=\left(x^{2}-1\right)^{\frac{1}{3}}$ and classify them as local max, min, or horizontal tangent.

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Problem 30488

Find the limit as $x$ approaches 1 for $\frac{\sin (\pi x)}{x-1}$ .

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Problem 30489

Find the derivative $f^{\prime}(x)$ for the function $f(x)=5 x\left(15 x^{4}+\frac{13}{x+1}\right)$ .

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Problem 30490

Find the limits: (a) $\lim_{x \rightarrow 2} f(x)$ , (b) $\lim_{x \rightarrow 0} g(x)$ , (c) $\lim_{x \rightarrow 2} g(f(x))$ .

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Problem 30491

Find the limit as $x$ approaches 3 for $\frac{x^{2}-4 x+3}{\sqrt{x+1}-\sqrt{5 x-11}}$ .

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Problem 30492

Find the tangent line equation for $f(x)=(3x-x^2)(3-x-x^2)$ at $x=1$ . $y=$

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Problem 30493

Find the velocity $v(t)=6 t^{2}-6 t-72$ and acceleration $a(t)=12 t-6$ of the particle. When is it slowing down or speeding up?

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Problem 30494

Calculate the average rate of change of $f(x)=x^{2}-3x+6$ from $x=1$ to $x=6$ .

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Problem 30495

Find the derivative of $f(x)=x^{2}+5$ using the definition $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

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Problem 30496

Which statement follows from: "If $f$ is continuous on $[a, b]$ , then $f$ has an absolute max and min on $[a, b]$ "?

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Problem 30497

Find $f^{\prime}(x)$ for $f(x)=-\frac{1}{3 x}$ and then calculate $f^{\prime}(2)$ .

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Problem 30498

Find $\frac{dy}{dx}$ using implicit differentiation for the equation $x^{7} = \cot y$ .

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Problem 30499

Graph $y=x^{2}-2 x-6$ and find the tangent line at the point where $x=-2$ .

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Problem 30500

Find the limits of the piecewise function $f(x)$ at $x = -2$ . Calculate: $\lim _{x \rightarrow-2^{-}} f(x), \lim _{x \rightarrow-2^{+}} f(x), \lim _{x \rightarrow-2^{2}} f(x).$

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Calculus

Problem 30401

Verkaufsprognose: Analysiere die Funktion f(t)=t3−24t2+150t+100f(t)=t^{3}-24 t^{2}+150 t+100f(t)=t3−24t2+150t+100 für Downloads in Tausend. Beantworte Fragen a) bis g).

Problem 30402

Find the rate of change of average cost when producing 179 belts, given C(x)=710+32x−0.062x2C(x)=710+32x-0.062x^{2}C(x)=710+32x−0.062x2.

Problem 30403

Find the rate of change of average cost when producing 179 belts, given C(x)=710+32x−0.062x2C(x)=710+32x-0.062x^{2}C(x)=710+32x−0.062x2. Calculate Cˉ′(x)\bar{C}^{\prime}(x)Cˉ′(x).

Problem 30404

At an excavation site, the dirt removed is modeled by f(t)f(t)f(t) where g(t)g(t)g(t) is given. (a) Find the average rate of change of fff from t=6t=6t=6 to t=12t=12t=12. (b) Approximate f′(9)f^{\prime}(9)f′(9) using the table data. Show your calculations.

Problem 30405

Is the function fff continuous for 0≤t≤120 \leq t \leq 120≤t≤12? Justify. Also, find f′(2)f^{\prime}(2)f′(2) for dirt removed at t=2t=2t=2 hours.

Problem 30406

The temperature T(t)=8tt2+4+98.6T(t)=\frac{8 t}{t^{2}+4}+98.6T(t)=t2+48t​+98.6 gives (a) the rate of change, (b) T(1)T(1)T(1), (c) change rate at t=1t=1t=1.

Problem 30407

Find the average cost change rate when producing 179 belts, given C(x)=750+35x−0.068x2C(x)=750+35x-0.068x^{2}C(x)=750+35x−0.068x2.

Problem 30408

Find the rate of change of average cost for producing 179 belts with C(x)=750+35x−0.068x2C(x)=750+35x-0.068x^{2}C(x)=750+35x−0.068x2.

Problem 30409

Find the temperature function T(t)=8tt2+4+98.6T(t)=\frac{8 t}{t^{2}+4}+98.6T(t)=t2+48t​+98.6. (a) Determine dTdt\frac{dT}{dt}dtdT​. (b) Calculate T(1)T(1)T(1). (c) Find dTdt\frac{dT}{dt}dtdT​ at t=1t=1t=1.

Problem 30410

Find the temperature function T(t)=8tt2+4+98.6T(t)=\frac{8t}{t^{2}+4}+98.6T(t)=t2+48t​+98.6. (a) Determine T′(t)T'(t)T′(t). (b) Calculate T(1)T(1)T(1). (c) Find T′(1)T'(1)T′(1).

Problem 30411

Find the derivative of y=x9x3y=\frac{x^{9}}{x^{3}}y=x3x9​ using the Quotient Rule and select the correct answer.

Problem 30412

Given the temperature function T(t)=8tt2+4+98.6T(t)=\frac{8 t}{t^{2}+4}+98.6T(t)=t2+48t​+98.6, find: (a) T′(t)T'(t)T′(t), (b) T(1)T(1)T(1), (c) T′(1)T'(1)T′(1).

Problem 30413

Find the derivative of y=x9x3y=\frac{x^{9}}{x^{3}}y=x3x9​ using the Quotient Rule and by dividing first.

Problem 30414

Find the derivative of y=x9x3 y=\frac{x^{9}}{x^{3}} y=x3x9​ using the Quotient Rule and select the correct answer.

Problem 30415

Divide x9x^{9}x9 by x3x^{3}x3, simplify to x6x^{6}x6, then find and simplify dydx=□\frac{d y}{d x}=\squaredxdy​=□.

Problem 30416

Problem 30417

Find the derivative of y=x9x3y=\frac{x^{9}}{x^{3}}y=x3x9​ using the Quotient Rule. Choose the correct derivative form from A, B, C, or D.

Problem 30418

Find the derivative of y=x6x4 y=\frac{x^{6}}{x^{4}} y=x4x6​ using the Quotient Rule and complete the answer choices.

Problem 30419

Cobalt-60 decays over time. Use the model A(t)=A0(12)t5.3A(t)=A_{0}\left(\frac{1}{2}\right)^{\frac{t}{5.3}}A(t)=A0​(21​)5.3t​ to answer:a) How long to reduce to half? b) Percent remaining after 10 years? c) Time until 10%10\%10% remains?

Problem 30420

Determinați dimensiunile unei săli de cinema cu volum de 4000 m³ pentru a minimiza f(x,y,z)=xy+2yz+2xzf(x, y, z) = xy + 2yz + 2xzf(x,y,z)=xy+2yz+2xz cu xyz=4000xyz = 4000xyz=4000. Folosiți metoda multiplicatorilor Lagrange.

Problem 30421

Find the derivative of y=x6x4y=\frac{x^{6}}{x^{4}}y=x4x6​ using the Quotient Rule and select the correct answer.

Problem 30422

Estimate the rate of change of f(x)=xx−4f(x)=\frac{x}{x-4}f(x)=x−4x​ at the point (2,−1)(2,-1)(2,−1).

Problem 30423

Find a formula for the sum ∑k=1n1(4k+3)(4k+7)\sum_{k=1}^{n} \frac{1}{(4 k+3)(4 k+7)}∑k=1n​(4k+3)(4k+7)1​ and compute ∑k=1∞1(4k+3)(4k+7)\sum_{k=1}^{\infty} \frac{1}{(4 k+3)(4 k+7)}∑k=1∞​(4k+3)(4k+7)1​.

Problem 30424

Divide x8x^{8}x8 by x3x^{3}x3, simplify to get x5x^{5}x5. Then find and simplify the derivative dydx\frac{d y}{d x}dxdy​.

Problem 30425

Find the acceleration of a particle with velocity v(t)=3+4.1cos⁡(0.9t)v(t)=3+4.1 \cos (0.9 t)v(t)=3+4.1cos(0.9t) at time t=4t=4t=4. Choices: (A) -2.016 (B) -0.677 (C) 1.633 (D) 1.814 (E) 2.978

Problem 30426

Divide x6x^{6}x6 by x4x^{4}x4, simplify to x2x^{2}x2. Find its derivative and verify dydx=□\frac{d y}{d x}=\squaredxdy​=□.

Problem 30427

Problem 30428

Find the limit: lim⁡x→0e2−cos⁡x−2xx2−2x\lim _{x \rightarrow 0} \frac{e^{2}-\cos x-2 x}{x^{2}-2 x}limx→0​x2−2xe2−cosx−2x​. Options: (A) −12-\frac{1}{2}−21​ (B) 0 (C) 12\frac{1}{2}21​ (D) 1 (E) nonexistent.

Problem 30429

Find the slope of the tangent line to x2+xy+y2=7x^{2}+x y+y^{2}=7x2+xy+y2=7 at the point (2,1)(2,1)(2,1). Choices: (A) −43-\frac{4}{3}−34​ (B) −54-\frac{5}{4}−45​ (C) -1 (D) −45-\frac{4}{5}−54​ (E) −34-\frac{3}{4}−43​.

Problem 30430

Find the values of ttt where the particle's position x(t)=t3−3t2−9t+1x(t)=t^{3}-3 t^{2}-9 t+1x(t)=t3−3t2−9t+1 is at rest.

Problem 30431

Divide x5/x3x^{5}/x^{3}x5/x3 and find the derivative of the result. Verify that dydx=□\frac{d y}{d x}=\squaredxdy​=□.

Problem 30432

Calculați integralele: 3) ∫01(x−1)3x4dx\int_{0}^{1} \frac{(x-1)^{3}}{\sqrt[4]{x}} d x∫01​4x​(x−1)3​dx; 4) ∫011−x2xdx\int_{0}^{1} \frac{1-x^{2}}{\sqrt{x}} d x∫01​x​1−x2​dx.

Problem 30433

Differentiate y=12x y=12 \sqrt{x} y=12x​. Find ddx(12x)=□ \frac{d}{d x}(12 \sqrt{x})=\square dxd​(12x​)=□.

Problem 30434

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=−10x8y=-\frac{10}{x^{8}}y=−x810​. What is dydx=?\frac{d y}{d x}=?dxdy​=?

Problem 30435

Differentiate the function: y=4x3−5x2+8x+9y=4 x^{3}-5 x^{2}+8 x+9y=4x3−5x2+8x+9. Find dydx\frac{d y}{d x}dxdy​.

Problem 30436

Find a formula for ∑k=1n(k+3)qk\sum_{k=1}^{n}(k+3) q^{k}∑k=1n​(k+3)qk with ∣q∣<1|q|<1∣q∣<1, and compute ∑k=1∞(k+3)qk\sum_{k=1}^{\infty}(k+3) q^{k}∑k=1∞​(k+3)qk.

Problem 30437

Find the derivative of the function: y=7x−52+8x−12+x5−8y=7 x^{-\frac{5}{2}}+8 x^{-\frac{1}{2}}+x^{5}-8y=7x−25​+8x−21​+x5−8. What is y′y'y′?

Problem 30438

Differentiate the function y=x11−11xy=\frac{x}{11}-\frac{11}{x}y=11x​−x11​. Find ddx(x11−11x)\frac{d}{dx}\left(\frac{x}{11}-\frac{11}{x}\right)dxd​(11x​−x11​).

Problem 30439

Find where the tangent line to y=−0.025x2+5xy=-0.025 x^{2}+5 xy=−0.025x2+5x has a slope of 9. The point(s) is/are □\square□.

Problem 30440

Differentiate the function y=4x2−32x3+7y=\frac{4 x^{2}-3}{2 x^{3}+7}y=2x3+74x2−3​. Find y′y^{\prime}y′.

Problem 30441

Verkaufsprognose: Analysiere die Funktion $f(t)=t^{3}-24 t^{2}+150 t+100$ für Downloads in Tausend. Beantworte Fragen a) bis g).

Find the rate of change of average cost when producing 179 belts, given $C(x)=710+32x-0.062x^{2}$ .

Find the rate of change of average cost when producing 179 belts, given $C(x)=710+32x-0.062x^{2}$ . Calculate $\bar{C}^{\prime}(x)$ .

At an excavation site, the dirt removed is modeled by $f(t)$ where $g(t)$ is given.
(a) Find the average rate of change of $f$ from $t=6$ to $t=12$ . (b) Approximate $f^{\prime}(9)$ using the table data. Show your calculations.

Is the function $f$ continuous for $0 \leq t \leq 12$ ? Justify. Also, find $f^{\prime}(2)$ for dirt removed at $t=2$ hours.

The temperature $T(t)=\frac{8 t}{t^{2}+4}+98.6$ gives (a) the rate of change, (b) $T(1)$ , (c) change rate at $t=1$ .

Find the average cost change rate when producing 179 belts, given $C(x)=750+35x-0.068x^{2}$ .

Find the rate of change of average cost for producing 179 belts with $C(x)=750+35x-0.068x^{2}$ .

Find the temperature function $T(t)=\frac{8 t}{t^{2}+4}+98.6$ . (a) Determine $\frac{dT}{dt}$ . (b) Calculate $T(1)$ . (c) Find $\frac{dT}{dt}$ at $t=1$ .

Find the temperature function $T(t)=\frac{8t}{t^{2}+4}+98.6$ . (a) Determine $T'(t)$ . (b) Calculate $T(1)$ . (c) Find $T'(1)$ .

Find the derivative of $y=\frac{x^{9}}{x^{3}}$ using the Quotient Rule and select the correct answer.

Given the temperature function $T(t)=\frac{8 t}{t^{2}+4}+98.6$ , find: (a) $T'(t)$ , (b) $T(1)$ , (c) $T'(1)$ .

Find the derivative of $y=\frac{x^{9}}{x^{3}}$ using the Quotient Rule and by dividing first.

Find the derivative of $y=\frac{x^{9}}{x^{3}}$ using the Quotient Rule and select the correct answer.

Divide $x^{9}$ by $x^{3}$ , simplify to $x^{6}$ , then find and simplify $\frac{d y}{d x}=\square$ .

Find the derivative of $y=\frac{x^{9}}{x^{3}}$ using the Quotient Rule. Choose the correct derivative form from A, B, C, or D.

Find the derivative of $y=\frac{x^{6}}{x^{4}}$ using the Quotient Rule and complete the answer choices.

Cobalt-60 decays over time. Use the model $A(t)=A_{0}\left(\frac{1}{2}\right)^{\frac{t}{5.3}}$ to answer:
a) How long to reduce to half? b) Percent remaining after 10 years? c) Time until $10\%$ remains?

Determinați dimensiunile unei săli de cinema cu volum de 4000 m³ pentru a minimiza $f(x, y, z) = xy + 2yz + 2xz$ cu $xyz = 4000$ . Folosiți metoda multiplicatorilor Lagrange.

Find the derivative of $y=\frac{x^{6}}{x^{4}}$ using the Quotient Rule and select the correct answer.

Estimate the rate of change of $f(x)=\frac{x}{x-4}$ at the point $(2,-1)$ .

Find a formula for the sum $\sum_{k=1}^{n} \frac{1}{(4 k+3)(4 k+7)}$ and compute $\sum_{k=1}^{\infty} \frac{1}{(4 k+3)(4 k+7)}$ .

Divide $x^{8}$ by $x^{3}$ , simplify to get $x^{5}$ . Then find and simplify the derivative $\frac{d y}{d x}$ .

Find the acceleration of a particle with velocity $v(t)=3+4.1 \cos (0.9 t)$ at time $t=4$ . Choices: (A) -2.016 (B) -0.677 (C) 1.633 (D) 1.814 (E) 2.978

Divide $x^{6}$ by $x^{4}$ , simplify to $x^{2}$ . Find its derivative and verify $\frac{d y}{d x}=\square$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{2}-\cos x-2 x}{x^{2}-2 x}$ . Options: (A) $-\frac{1}{2}$ (B) 0 (C) $\frac{1}{2}$ (D) 1 (E) nonexistent.

Find the slope of the tangent line to $x^{2}+x y+y^{2}=7$ at the point $(2,1)$ . Choices: (A) $-\frac{4}{3}$ (B) $-\frac{5}{4}$ (C) -1 (D) $-\frac{4}{5}$ (E) $-\frac{3}{4}$ .

Find the values of $t$ where the particle's position $x(t)=t^{3}-3 t^{2}-9 t+1$ is at rest.

Divide $x^{5}/x^{3}$ and find the derivative of the result. Verify that $\frac{d y}{d x}=\square$ .

Calculați integralele: 3) $\int_{0}^{1} \frac{(x-1)^{3}}{\sqrt[4]{x}} d x$ ; 4) $\int_{0}^{1} \frac{1-x^{2}}{\sqrt{x}} d x$ .

Differentiate $y=12 \sqrt{x}$ . Find $\frac{d}{d x}(12 \sqrt{x})=\square$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=-\frac{10}{x^{8}}$ . What is $\frac{d y}{d x}=?$

Differentiate the function: $y=4 x^{3}-5 x^{2}+8 x+9$ . Find $\frac{d y}{d x}$ .

Find a formula for $\sum_{k=1}^{n}(k+3) q^{k}$ with $|q|<1$ , and compute $\sum_{k=1}^{\infty}(k+3) q^{k}$ .

Find the derivative of the function: $y=7 x^{-\frac{5}{2}}+8 x^{-\frac{1}{2}}+x^{5}-8$ . What is $y'$ ?

Differentiate the function $y=\frac{x}{11}-\frac{11}{x}$ . Find $\frac{d}{dx}\left(\frac{x}{11}-\frac{11}{x}\right)$ .

Find where the tangent line to $y=-0.025 x^{2}+5 x$ has a slope of 9. The point(s) is/are $\square$ .

Differentiate the function $y=\frac{4 x^{2}-3}{2 x^{3}+7}$ . Find $y^{\prime}$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2 x^{2}-4 x+1$ and calculate $f^{\prime}(-1)$ .

Find the derivative of $f(x)=(x-7)(2x+5)$ using the Product Rule and by expanding the product.

Given the equation $R(v)=\frac{6400}{v}$ , find: a) $R'(v)$ , b) $R(90)$ , c) $R'(90)$ . Round as needed.

Differentiate the function $F(x)=(x^{5}-8 x)^{2}$ . Find $F^{\prime}(x)=\square$ .

Find the rate of change of average revenue when 400 jackets are produced, given $R(x)=85 \sqrt{x}$ .

Differentiate the function $y=\sqrt{x^{2}+11}$ . Find $\frac{d y}{d x}=\square$ .

Find the derivative of the function $f(x) = 2 e^{x} - \frac{1}{4} x^{4}$ . What is $f^{\prime}(x)$ ?

Differentiate the function $f(x)=\sqrt{x}-(x-1)^{3}$ . Find $f^{\prime}(x)=$ .

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}+x+7}-3}{x^{2}-6 x+5}$ .

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

Berechne $U_{n}$ und $U_{\underline{O}}$ für $f(x)=x^{2}$ im Intervall $[0 ; 1]$ , unterteilt in $n$ Teilintervalle.

Differentiate the function $f(x)=\left(\frac{x-4}{x+3}\right)^{6}$ . Find $f^{\prime}(x)=$ .

A student learns $N(t)=20t-t^{2}$ terms after $t$ hours. Find terms learned from $t=2$ to $t=3$ and rate at $t=2$ .

Find the first and second derivatives of the function $f(x) = 3 e^{x} + 2 x^{3} - x + 5$ .

Differentiate the function $f(x)=(2 x^{5}-5 x^{3}+3)^{26}$ . Find $f^{\prime}(x)=$ .

Find the first and second derivatives of the function $f(x) = \frac{2}{x} - 3 e^{x}$ .

Find the first and second derivatives of the function $f(x) = 0.3 e^{x} - \sqrt{x}$ .

Find the rate of change of the particle's position given $s=\sqrt{45+6t}$ at $t=6 \mathrm{sec}$ . Answer: $\square \mathrm{m/sec}$ .

Find $y^{\prime}$ if $y=\frac{x \ln (x)}{1+\ln (x)}$ . Choose from options a, b, c, d, or e.

Find the derivative $\frac{d C}{d x}$ of the function $C(x)=9.16 x^{4}-85.05 x^{3}+287.45 x^{2}-309.42 x+2651.5$ . Interpret its meaning.

Gegeben ist die Funktion $f_k(x) = x^2 + 2kx + 2x + 1$ . Finde die Tiefpunkte und deren Koordinaten.

Find the second derivative $f^{\prime \prime}(x)$ for the function $f(x)=3 x^{2}-11 x-\frac{5}{x^{2}}$ .

Model consumer credit with $C(x)=9.16 x^{4}-85.05 x^{3}+287.45 x^{2}-309.42 x+2651.5$ . Estimate rise in 2015.

Given $s(t)=5 t^{2}+20 t$ , find: a) $v(t)$ , b) $a(t)$ , c) velocity and acceleration at $t=2$ sec.

A medicine's amount $M(t)=-\frac{1}{3} t^{2}+t$ in blood changes over time. Find $M'(2)$ and explain if it's negative.

An object falls from rest; find (a) distance $s(5)$ , (b) speed $v(5)$ , and (c) acceleration after $5$ seconds.

Find the rate of change of temperature $T(h)=\frac{60}{h+2}$ with respect to height at $h=3$ km.

Find the second derivative $f_{k}^{\prime \prime}(-k-1)$ for $f_{k}(x) = x^2 + 2kx + 2x + 1$ .

Calculate the Riemann sum $R_{n}$ for $f(x)=-\frac{x^{2}}{3}-2$ on $[0,3]$ as a function of $n$ . Find $\lim_{n \to \infty} R_{n}$ .