Calculus

Problem 30501

Find the derivative of the function $h(z) = 27z - z^3$ .

See Solution

Problem 30502

Find the extreme values of the function $y=x^{3}-12 x+2$ and identify where they occur.

See Solution

Problem 30503

Find the tangent line equation for the curve $f(x)=x^{2}+11 x-15$ at $x=1$ .

See Solution

Problem 30504

Find the tangent line equation for the function $f(x)=2 x^{2}-3 x+6$ at the point where $x=2$ .

See Solution

Problem 30505

Determine where the function $h(z) = 27z - z^3$ is increasing or decreasing on the intervals given.

See Solution

Problem 30506

Find $c$ such that $\frac{f(-1)-f(-2)}{-1-(-2)}=f(c)$ for $f(x)=x^{2}+5x+3$ in $[-2,-1]$ .

See Solution

Problem 30507

Find $c$ such that $\frac{f(8)-f(4)}{8-4}=f^{\prime}(c)$ for $f(x)=\ln (x-3)$ on $[4,8]$ . Round to the nearest thousandth.

See Solution

Problem 30508

Find the half-life of a drug if the initial level is $3,600 \mathrm{mg} / \mathrm{L}$ and after one day it is $1,160 \mathrm{mg} / \mathrm{L}$ .

See Solution

Problem 30509

1. Two people walk away from the same point at $3 \mathrm{mi/h}$ and $4 \mathrm{mi/h}$ . Find the distance change rate after 15 min.
2. Given $\mathrm{PV}^{1.4}=\mathrm{C}$ , find the volume increase rate when $V=400 \mathrm{cm}$ and $P=80 \mathrm{kPa}$ , $dP/dt=-10 \mathrm{kPa/min}$ .
3. A trough is $8 \mathrm{ft}$ long with triangular ends. Water fills at $10 \mathrm{ft}^3/min$ . Find the water level rise rate at 6 inches deep.
4. Gravel forms a cone with height and diameter equal, dumped at $30 \mathrm{ft}^3/min$ . Find height increase rate when $h=10 \mathrm{ft}$ .
5. A person is 15 m from a building, watching an elevator. When the angle of elevation is 1 rad and changing at 0.2 rad/sec, find elevator speed.
6. A plane flies at $5 \mathrm{km}$ altitude. When the angle of elevation is $\frac{\pi}{3}$ and decreasing at $\frac{\pi}{6} \mathrm{rad/min}$ , find the plane's speed.

See Solution

Problem 30510

Find the intervals where the function $f(x)=x^{3}+3 x^{2}-x-24$ is concave up or down.

See Solution

Problem 30511

Calculate the average rate of change of $f(x)=3 x^{2}-1$ over the interval $[-4,4]$ .

See Solution

Problem 30512

Water is wetting a circular area expanding at $5 \text{ mm}^2/\text{s}$ . Find the radius expansion rate at $155 \text{ mm}$ .

See Solution

Problem 30513

Find the tangent line equation at the point (0,1) for the curve $y^{4}+x^{3}=y^{2}+12 x$ .

See Solution

Problem 30514

1. Two people walk in different directions. Find the rate of distance change after 15 min when one walks $3 \mathrm{mi/h}$ and the other $4 \mathrm{mi/h}$ northeast.
2. Given $PV^{1.4}=C$ , find the volume increase rate when $V=400 \mathrm{cm}$ , $P=80 \mathrm{kPa}$ , and $dP/dt=-10 \mathrm{kPa/min}$ .
3. A trough is $8 \mathrm{ft}$ long with triangular ends. Find the water level rise rate when filled at $10 \mathrm{ft^3/min}$ and water is $6$ inches deep.
4. Gravel forms a cone shape with height equal to diameter. Find height increase rate when the pile is $10 \mathrm{ft}$ high and gravel is dumped at $30 \mathrm{ft^3/min}$ .
5. A person is 15 meters from a building watching an elevator. Find elevator speed when angle of elevation is $1$ rad and changing at $0.2 \mathrm{rad/sec}$ .
6. A plane flies at $5 \mathrm{km}$ altitude. Find speed when angle of elevation is $\frac{\pi}{3}$ and decreasing at $\frac{\pi}{6} \mathrm{rad/min}$ .

See Solution

Problem 30515

Find the derivative of $f(x)=\frac{9 e^{x}}{2 e^{x}+1}$ .

See Solution

Problem 30516

Find the derivative of $y$ where $y=3 \sin^{-1}(5 x^{4})$ .

See Solution

Problem 30517

Find values of $c$ for $f(x)=x+\frac{45}{x}$ in $[5,9]$ that satisfy $\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$ .

See Solution

Problem 30518

1. For adiabatic expansion, given $PV^{1.4}=C$ , find the volume increase rate when $V=400 \mathrm{cm}$ , $P=80 \mathrm{kPa}$ , and $dP/dt=-10 \mathrm{kPa/min}$ .
2. A trough is 8 ft long with triangular ends (2 ft wide, 1 ft high). If water fills at $10 \mathrm{ft}^3/min$ , find the water level rise rate at 6 inches deep.
3. Gravel forms a conical pile with base diameter equal to height, dumped at $30 \mathrm{ft}^3/min$ . Find height increase rate when pile is 10 ft high.
4. A person 15 m from a building observes an elevator. When the angle of elevation is 1 rad and changing at 0.2 rad/sec, find the elevator speed.
5. A plane at 5 km altitude passes over a telescope. When angle of elevation is $\frac{\pi}{3}$ and decreasing at $\frac{\pi}{6} \mathrm{rad/min}$ , find the plane's speed.

See Solution

Problem 30519

Determine the intervals where the function $G(x)=\frac{1}{x^{2}}+7$ is increasing or decreasing.

See Solution

Problem 30520

A ladder 20 ft long slips down a wall at 2 ft/s. How fast is the bottom moving when it's 16 ft from the wall?

See Solution

Problem 30521

Find critical points and the max/min values of $f(x)=x^{3}-12x-3$ on the interval $I=[-3,5]$ .

See Solution

Problem 30522

Find critical points and determine the max and min of $f(x)=x^{3}-12 x-3$ on the interval $I=[-3,5]$ .

See Solution

Problem 30523

Find the derivative of $y$ with respect to $x$ , $t$ , or $\theta$ : $y=\ln(6x)$ .

See Solution

Problem 30524

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

See Solution

Problem 30525

Determine where the function $G(x)=\frac{1}{4} x^{4}-x^{3}+15$ is concave up, concave down, and find inflection points.

See Solution

Problem 30526

Find the derivative of $y = x^{5} \ln x - \frac{1}{3} x^{3}$ with respect to $x$ .

See Solution

Problem 30527

1) For $f(x)=\sqrt{x-7}$ : a) Find the domain and sketch. b) Average rate of change on $[7, 10]$ . c) Estimate IRC at $x=8$ . d) Verify IRC at $x=8$ using limits.
2) For $s(t)=125-5t^2$ : a) Time to hit ground. b) Average velocity: i) total fall, ii) fourth second. c) Estimate velocity at impact. d) Velocity at $t=4$ using limits.
3) For $y=2\sin(2x)$ on $[0, \pi]$ : a) $x$ where IRC is positive and $y<0$ . b) $x$ where IRC is negative and $y<0$ , and where IRC is zero and $y>0$ . c) Estimate IRC from part a) using a small $h$ .

See Solution

Problem 30528

Find the limit as $x$ approaches 2 from the right for the expression $3x + 1$ .

See Solution

Problem 30529

Find the slope of secant lines for $f(x)=(x-1)^{1/3}$ between $(1,0)$ and $(x,f(x))$ at $x=0.5,0.9,0.99,0.9999$ .

See Solution

Problem 30530

An object's height is $s(t)=125-5t^{2}$ . Find: a) time to hit ground, b) average velocity, c) velocity at impact, d) velocity at $t=4$ . For $y=2\sin 2x$ , find $x$ where: a) rate positive, $y$ negative, b) rate negative, $y$ negative, c) rate zero, $y$ positive.

See Solution

Problem 30531

Find $f^{\prime \prime}(x)$ using the chain rule for $f^{\prime \prime}(x)=\frac{d}{d x}\left(\frac{2 x}{3\left(x^{2}-1\right)^{2 / 3}}\right)$ .

See Solution

Problem 30532

An object's height is $s(t)=125-5t^{2}$ . Find when it hits the ground, average velocity, and velocity at $t=4$ .

See Solution

Problem 30533

Find the limit as $x$ approaches 3 from the right: $\lim _{x \rightarrow 3^{+}}\left(\frac{x+2}{x-1}\right)$ .

See Solution

Problem 30534

Find the limit as $x$ approaches -1 from the left for the expression $1 - 5x$ .

See Solution

Problem 30535

Find the average rate of change of $f(x)=\frac{6}{3 x-7}$ on $[-3,0]$ . What is the average rate of change?

See Solution

Problem 30536

A trough is 8 ft long, 2 ft wide, and 1 ft high. Water fills at 10 ft³/min. How fast is the water level rising at 6 in deep?

See Solution

Problem 30537

Find the midpoint Riemann sum for $\int_{2}^{3.5} x^{3} dx$ using 6 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 30538

Find the function $f(x)$ given $f^{\prime \prime}(x)=2 x$ , $f^{\prime}(2)=-1$ , and $f(3)=1$ .

See Solution

Problem 30539

Find the function $f(x)$ given $f^{\prime \prime}(x)=2$ , $f^{\prime}(1)=4$ , and $f(2)=-2$ .

See Solution

Problem 30540

Vérifiez si $F$ est une primitive de $f$ sur $K$ pour les cas suivants : a) $F(x)=8 x^{3}-12 x^{2}+6 x-7$ , $f(x)=6(2 x-1)^{2}$ , $K=R$ b) $F(x)=-2 \cos (3 x+2)$ , $f(x)=6 \sin (3 x+2)$ , $K=R$ c) $F(x)=\sqrt{2 x+1}$ , $f(x)=\frac{1}{\sqrt{2 x+1}}$ , $K=\left\lvert\,-\frac{1}{2}\right. ;+\infty[$ d) $F(x)=\left(\frac{1}{x}-\sqrt{x}\right)^{2}$ , $f(x)=\frac{(x \sqrt{x}-1)(2+x \sqrt{x})}{x^{3}}$ , $K=10 ;+\infty[$

See Solution

Problem 30541

Find the function $f(x)$ given $f^{\prime \prime}(x)=\frac{1}{x^{3/2}}$ , $f^{\prime}(4)=2$ , and $f(0)=1$ .

See Solution

Problem 30542

Find the slope of secant lines for $f(x)=(x-1)^{1 / 3}$ between $(1,0)$ and $(x,f(x))$ at $x=1.5,1.1,1.01,1.0001$ .

See Solution

Problem 30543

Find the limits: $\lim_{x \to 4^-} f(x)$ , $\lim_{x \to 4^+} f(x)$ , and $\lim_{x \to 4} f(x)$ for the piecewise function $f(x)$ .

See Solution

Problem 30544

Show the derivative of the loss function $E(\mathbf{w})=-\sum_{n=1}^{N} \sum_{k=1}^{K} t_{k n} \ln y_{k}(\mathbf{x}_{n}, \mathbf{w})$ with respect to $a_{k}$ , using $\frac{\partial y_{k}}{\partial a_{j}}=y_{k}(I_{k j}-y_{j})$ and $y_{k}(\phi)=\frac{\exp(a_{k})}{\sum_{j} \exp(a_{j})}$ .

See Solution

Problem 30545

Déterminez une primitive de $f$ sur l'intervalle $K$ pour chaque cas : a) $f(x)=4 x^{3}-5 x^{2}-1$ , $K=\mathbb{R}$ ; b) $f(x)=\frac{1}{x^{2}}+\frac{1}{x^{3}}$ , $K=1-\infty ; 0$ ; c) $f(x)=x+\frac{1}{2 \sqrt{x}}$ , $K=10 ;+\infty$ ; d) $f(x)=\cos x-2 \sin x$ , $K=\mathbb{R}$ .

See Solution

Problem 30546

Show that the derivative of the loss function $E(\mathbf{w})$ satisfies $\frac{\partial y_{k}}{\partial a_{j}}=y_{k}(\delta_{kj}-y_{j})$ for softmax output units.

See Solution

Problem 30547

What do the secant lines indicate about the tangent line of the function $f(x)=(x-1)^{1/3}$ at $x=1$ ?

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Problem 30548

Show the derivative of the loss function $E(\mathbf{w})=-\sum_{n=1}^{N} \sum_{k=1}^{K} t_{k n} \ln y_{k}(\mathbf{x}_{n}, \mathbf{w})$ with respect to the activation $a_{k}$ for softmax, satisfying $\frac{\partial y_{k}}{\partial a_{j}}=y_{k}(I_{k j}-y_{j})$ .

See Solution

Problem 30549

Find the limit: $\lim _{h \rightarrow 1} \frac{h^{2}+3 h-4}{h-1}$ . Options: A. 0, B. -3, C. 5, D. undefined.

See Solution

Problem 30550

Find the limit as $h$ approaches 0 of $\frac{f(x+h)-f(x)}{h} = \frac{2(x+h)\sqrt{2(x+h)+3} - 2x\sqrt{2x+3}}{h}$ .

See Solution

Problem 30551

Evaluate and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=7 x^{4}+x-8$ without remaining division by $h$ .

See Solution

Problem 30552

Find the radius $r$ and height $h$ of a cylindrical can using $1536 \pi$ in² of metal for maximum volume.

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Problem 30553

Déterminez une primitive de $f$ sur $\mathbb{R}$ pour chaque cas : a) $f(x)=5 x(5 x^{2}-7)^{4}$ , b) $f(x)=\frac{2 x-3}{(2 x^{2}-6 x+11)^{3}}$ , c) $f(x)=\frac{x(2 x^{2}+1)}{\sqrt{x^{4}+x^{2}+1}}$ , d) $f(x)=3 \sin(3 x-2)$ , e) $f(x)=2 \cos x \sin^{4} x$ et $f(x)=x \cos(3 x^{2}-\frac{\pi}{4})$ .
Pour 1.d, trouvez $\mathrm{F}$ de $f$ sur $\mathrm{K}$ avec conditions : a) $f(x)=x^{3}-\frac{2}{x^{2}}$ , $K=] 0 ;+\infty[$ , $F(2)=0$ ; b) $f(x)=3 \sin x-4 \cos x$ , $K=\mathbb{R}$ , $F(\pi)=-1$ ; c) $f(x)=2 x-\frac{1}{x^{2}}-\frac{1}{\sqrt{x}}$ , $K=] 0 ;+\infty[$ , $F(1)=1$ ; d) $f(x)=-\frac{1}{\cos^{2} x}+\sin x$ , $K=-\frac{\pi}{2} ; \frac{\pi}{2}[$ , $F(0)=1$ .

See Solution

Problem 30554

Calculate the account value after 11 years for an investment of \ $3950 at a 9.9% annual rate, using$ V=P e^{r t}$.

See Solution

Problem 30555

Evaluate the integral: $\int \frac{\sqrt{x^{3}}-\sqrt[3]{x}}{3 \sqrt[4]{x}} d x$

See Solution

Problem 30556

Find the integral: $\int \frac{\sqrt{x^{3}}-\sqrt[3]{x}}{3 \sqrt[4]{x}} d x$

See Solution

Problem 30557

Evaluate the function $f(x)=-x^{3}+5 x^{2}+2 x+5$ for: a. $f(x+h)$ b. $\frac{f(x+h)-f(x)}{h}$ (no $h$ in the answer).

See Solution

Problem 30558

Find the acceleration due to gravity at a distance of $1.83 \times 10^{8} \mathrm{~m}$ above Earth's surface. Units: $\mathrm{m} / \mathrm{s}^{\wedge} 2$ .

See Solution

Problem 30559

What is the robin's displacement $\Delta x$ from $t=0$ s to $t=1.5$ s based on its velocity graph? Answer with two significant digits.

See Solution

Problem 30560

When does a 15m falling object, with $s(t)=15-6.729 t^{2}$ , hit the ground? Round $t$ to 3 decimal places.

See Solution

Problem 30561

Create a second-order ODE with a twice-differentiable function $y(x)$ . Verify a solution by following these steps:
1. Compute $y'(x)$ and $y''(x)$ .
2. Choose constants $a_2(x)$ and $a_1(x)$ , then calculate $f(x) = a_2(x) y''(x) + a_1(x) y'(x)$ .
3. Show $y_C(x) = y(x) + C$ is a solution of $f(x) = 0$ .
4. Find $C$ such that $y_C(x_0) = k$ for chosen $x_0$ and $k$ . Solve the initial value problem (IVP).
5. Construct a different second-order IVP and verify the solution using the Fundamental Theorem of Calculus (FTOC).

See Solution

Problem 30562

Find $y^{\prime}(e)$ for the function $y=e^{x-e}+\ln x-\frac{x}{e}$ .

See Solution

Problem 30563

Find the average velocity of an object falling from 15 m with $s(t)=15-6.729 t^{2}$ after $t$ seconds (round to 3 decimals).

See Solution

Problem 30564

Find the derivative of $F(t) = \ln(\sinh(3t))$ . What is $F'(t)$ ?

See Solution

Problem 30565

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

See Solution

Problem 30566

Find the derivative of $g(x)=\sinh ^{2}(x)$ . What is $g^{\prime}(x)=?$

See Solution

Problem 30567

Find the velocity of an object falling from 15 meters, given $s(t)=15-6.729 t^{2}$ , when it hits the ground.

See Solution

Problem 30568

Find the limit: $\lim _{x \rightarrow \infty} x \cdot \sin \left(\frac{\pi}{x}\right)$ .

See Solution

Problem 30569

Find $a$ if $\lim _{x \rightarrow-\infty} \frac{\sqrt{a x^{2}+b x+c}}{d-a x}=2$ .

See Solution

Problem 30570

Find the derivative $\frac{d y}{d x}$ for the equation $y=\log _{x} e$ .

See Solution

Problem 30571

Find the average rate of change of $f(x)=\left(\frac{1}{2}\right)^{x}$ from $x=-2$ to $x=2$ .

See Solution

Problem 30572

Evaluate the integral $\int_{-4}^{1} \sqrt{2-x} \, \mathrm{d} x$ and round to 3 significant figures.

See Solution

Problem 30573

Find the function $f(x)$ if $f^{\prime}(x)=4-6 x$ and $f(1)=0$ .

See Solution

Problem 30574

Find the derivative of $\sqrt{\frac{x-3}{2x+5}}$ using quotient and chain rule, showing all steps.

See Solution

Problem 30575

Find the derivative of the function $y=(x+4)(x+2)$ . Calculate $\frac{d y}{d x}$ .

See Solution

Problem 30576

Find the point on the curve $y=\cosh (x)$ where the tangent slope equals 5. $(x, y)=(\square)$

See Solution

Problem 30577

Find the limit: $L = \lim _{x \rightarrow \frac{\pi}{4}} \frac{\tan x - 1}{x - \frac{\pi}{4}}$ .

See Solution

Problem 30578

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

See Solution

Problem 30579

Find $\frac{d y}{d x}$ for the equation $y=5 x^{2}-3 x+7$ at $x=1$ .

See Solution

Problem 30580

Find the derivative $\frac{d y}{d x}$ for the curve $y=x^{4}-3 x^{3}+x$ and the tangent at $x=-1$ .

See Solution

Problem 30581

Find $\frac{d^{2} h}{d t^{2}}$ given $\frac{d h}{d t}=-\sqrt{5 h}$ at $h=16$ .

See Solution

Problem 30582

Find the derivative of $y=2 x^{2}-3 x+1$ and the tangent line $L$ at the point $(3,10)$ .

See Solution

Problem 30583

Find the derivative of $f(x) = x^{2}(x+3)^{4}$ using product and chain rule, and show your work.

See Solution

Problem 30584

Differentiate the function: $\frac{5+2 \sqrt{x}}{(5-4 x)^{3}}$ .

See Solution

Problem 30585

Find $m'(2)$ for $m(x) = \frac{g(x)}{f(x)}$ using given values and the quotient rule.

See Solution

Problem 30586

Sketch the graph of $P(t)=\frac{1200}{\sqrt{2030-t}}$ for $t$ from 2000 to 2030. Find $\lim_{t \rightarrow 2030^{-}} P(t)$ and discuss the model's validity near $t=2030$ .

See Solution

Problem 30587

Find the derivative of $f(x) = x \sqrt{x^{2}+1}$ using the product and chain rule, showing all steps.

See Solution

Problem 30588

Find where the tangent line to $y=\frac{\csc ^{2}\left(\frac{x}{2}\right)}{3 x+4}$ is horizontal for $0 \leq x \leq 2 \pi$ .

See Solution

Problem 30589

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

See Solution

Problem 30590

Find the sixth derivative of the function $f(x) = x^{6}$ .

See Solution

Problem 30591

Find the fifth derivative of the function $f(x)=5 x^{7}+20 x^{6}+14 x^{5}+8 x^{4}+3 x^{3}$ .

See Solution

Problem 30592

Find the antiderivative $F(x)$ of $f(x)=\frac{8}{x^{2}}-\frac{10}{x^{5}}$ with $F(1)=0$ . What is $F(x)$ ?

See Solution

Problem 30593

1. Find the average rate of change of $f(x)=x^{3}-2x^{2}+7$ from $x=-1$ to $x=3$ .
2. Who's crystal grew faster: Kristin's (0.1g to 5g in 3 days) or Husain's (0.1g to 15g in 10 days)?
3. Estimate the submarine's instantaneous depth change at $t=3s$ from given depth data.
4. Estimate the instantaneous rate of change of $f(x)=2^{x}-2x+1$ at $x=-1$ .
5. Estimate the slope of the tangent at $x=2$ from the given graph.
6. Which distance vs. time graph fits an athlete's run with varying lap speeds?
7. At $x=5$ , does $f(x)=13x-1.3x^{2}+7.3$ have a max, min, both, or neither?
8. What is the difference quotient for $f(x)=2x^{2}-3x+9$ on $3 \leq x \leq 3+h$ ?
9. What is the maximum value of $y=35(1.7)^{x}$ for $0 \leq x \leq 8$ ?
10. Which of the following is not a polynomial function?
11. Which type of polynomial cannot be represented by the given graph?

See Solution

Problem 30594

Find the antiderivative $F(x)$ of $f(x)=5 \cos x - 7 \sin x$ with $F(0)=7$ .

See Solution

Problem 30595

Find the second derivative $D_{x}^{2} y$ for the function $y=\frac{1-x}{x-3}$ .

See Solution

Problem 30596

Find the general antiderivative of $f(x)=3 e^{x}+8 \sec ^{2} x$ . Answer: $F(x)=$ with constant $C$ .

See Solution

Problem 30597

Find the derivative of $y=\sin ^{2}(x^{3})$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 30598

Find the antiderivative $F(x)$ of $f(x)=x^{3}+4 \sqrt{x}$ given $F(1)=-8$ . What is $F(x)$ ?

See Solution

Problem 30599

Find the antiderivative $F(x)$ of $f(x)=5 \cos x-7 \sin x$ with $F(0)=7$ . What is $F(x)$ ?

See Solution

Problem 30600

Find the third derivative $\frac{d^{3} y}{d x^{3}}$ for the function $y=\frac{2}{x^{2}-3}$ .

See Solution

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Calculus

Problem 30501

Find the derivative of the function h(z)=27z−z3h(z) = 27z - z^3h(z)=27z−z3.

Problem 30502

Find the extreme values of the function y=x3−12x+2y=x^{3}-12 x+2y=x3−12x+2 and identify where they occur.

Problem 30503

Find the tangent line equation for the curve f(x)=x2+11x−15f(x)=x^{2}+11 x-15f(x)=x2+11x−15 at x=1x=1x=1.

Problem 30504

Find the tangent line equation for the function f(x)=2x2−3x+6f(x)=2 x^{2}-3 x+6f(x)=2x2−3x+6 at the point where x=2x=2x=2.

Problem 30505

Determine where the function h(z)=27z−z3h(z) = 27z - z^3h(z)=27z−z3 is increasing or decreasing on the intervals given.

Problem 30506

Find ccc such that f(−1)−f(−2)−1−(−2)=f(c)\frac{f(-1)-f(-2)}{-1-(-2)}=f(c)−1−(−2)f(−1)−f(−2)​=f(c) for f(x)=x2+5x+3f(x)=x^{2}+5x+3f(x)=x2+5x+3 in [−2,−1][-2,-1][−2,−1].

Problem 30507

Find ccc such that f(8)−f(4)8−4=f′(c)\frac{f(8)-f(4)}{8-4}=f^{\prime}(c)8−4f(8)−f(4)​=f′(c) for f(x)=ln⁡(x−3)f(x)=\ln (x-3)f(x)=ln(x−3) on [4,8][4,8][4,8]. Round to the nearest thousandth.

Problem 30508

Find the half-life of a drug if the initial level is 3,600mg/L3,600 \mathrm{mg} / \mathrm{L}3,600mg/L and after one day it is 1,160mg/L1,160 \mathrm{mg} / \mathrm{L}1,160mg/L.

Problem 30509

Problem 30510

Find the intervals where the function f(x)=x3+3x2−x−24f(x)=x^{3}+3 x^{2}-x-24f(x)=x3+3x2−x−24 is concave up or down.

Problem 30511

Calculate the average rate of change of f(x)=3x2−1f(x)=3 x^{2}-1f(x)=3x2−1 over the interval [−4,4][-4,4][−4,4].

Problem 30512

Water is wetting a circular area expanding at 5 mm2/s5 \text{ mm}^2/\text{s}5 mm2/s. Find the radius expansion rate at 155 mm155 \text{ mm}155 mm.

Problem 30513

Find the tangent line equation at the point (0,1) for the curve y4+x3=y2+12xy^{4}+x^{3}=y^{2}+12 xy4+x3=y2+12x.

Problem 30514

Problem 30515

Find the derivative of f(x)=9ex2ex+1f(x)=\frac{9 e^{x}}{2 e^{x}+1}f(x)=2ex+19ex​.

Problem 30516

Find the derivative of yyy where y=3sin⁡−1(5x4)y=3 \sin^{-1}(5 x^{4})y=3sin−1(5x4).

Problem 30517

Find values of ccc for f(x)=x+45xf(x)=x+\frac{45}{x}f(x)=x+x45​ in [5,9][5,9][5,9] that satisfy f(b)−f(a)b−a=f′(c)\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)b−af(b)−f(a)​=f′(c).

Problem 30518

Problem 30519

Determine the intervals where the function G(x)=1x2+7G(x)=\frac{1}{x^{2}}+7G(x)=x21​+7 is increasing or decreasing.

Problem 30520

A ladder 20 ft long slips down a wall at 2 ft/s. How fast is the bottom moving when it's 16 ft from the wall?

Problem 30521

Find critical points and the max/min values of f(x)=x3−12x−3f(x)=x^{3}-12x-3f(x)=x3−12x−3 on the interval I=[−3,5]I=[-3,5]I=[−3,5].

Problem 30522

Find critical points and determine the max and min of f(x)=x3−12x−3f(x)=x^{3}-12 x-3f(x)=x3−12x−3 on the interval I=[−3,5]I=[-3,5]I=[−3,5].

Problem 30523

Find the derivative of yyy with respect to xxx, ttt, or θ\thetaθ: y=ln⁡(6x)y=\ln(6x)y=ln(6x).

Problem 30524

Find the tangent line equation to the curve f(x)=e2xf(x)=e^{2 x}f(x)=e2x at the point (0,f(0))(0, f(0))(0,f(0)).

Problem 30525

Determine where the function G(x)=14x4−x3+15G(x)=\frac{1}{4} x^{4}-x^{3}+15G(x)=41​x4−x3+15 is concave up, concave down, and find inflection points.

Problem 30526

Find the derivative of y=x5ln⁡x−13x3y = x^{5} \ln x - \frac{1}{3} x^{3}y=x5lnx−31​x3 with respect to xxx.

Problem 30527

Problem 30528

Find the limit as xxx approaches 2 from the right for the expression 3x+13x + 13x+1.

Problem 30529

Find the slope of secant lines for f(x)=(x−1)1/3f(x)=(x-1)^{1/3}f(x)=(x−1)1/3 between (1,0)(1,0)(1,0) and (x,f(x))(x,f(x))(x,f(x)) at x=0.5,0.9,0.99,0.9999x=0.5,0.9,0.99,0.9999x=0.5,0.9,0.99,0.9999.

Problem 30530

Problem 30531

Find f′′(x)f^{\prime \prime}(x)f′′(x) using the chain rule for f′′(x)=ddx(2x3(x2−1)2/3)f^{\prime \prime}(x)=\frac{d}{d x}\left(\frac{2 x}{3\left(x^{2}-1\right)^{2 / 3}}\right)f′′(x)=dxd​(3(x2−1)2/32x​).

Problem 30532

An object's height is s(t)=125−5t2s(t)=125-5t^{2}s(t)=125−5t2. Find when it hits the ground, average velocity, and velocity at t=4t=4t=4.

Problem 30533

Find the limit as xxx approaches 3 from the right: lim⁡x→3+(x+2x−1)\lim _{x \rightarrow 3^{+}}\left(\frac{x+2}{x-1}\right)limx→3+​(x−1x+2​).

Problem 30534

Find the limit as xxx approaches -1 from the left for the expression 1−5x1 - 5x1−5x.

Problem 30535

Find the average rate of change of f(x)=63x−7f(x)=\frac{6}{3 x-7}f(x)=3x−76​ on [−3,0][-3,0][−3,0]. What is the average rate of change?

Problem 30536

A trough is 8 ft long, 2 ft wide, and 1 ft high. Water fills at 10 ft³/min. How fast is the water level rising at 6 in deep?

Problem 30537

Find the midpoint Riemann sum for ∫23.5x3dx\int_{2}^{3.5} x^{3} dx∫23.5​x3dx using 6 equal subintervals. Round to the nearest thousandth.

Problem 30538

Find the function f(x)f(x)f(x) given f′′(x)=2xf^{\prime \prime}(x)=2 xf′′(x)=2x, f′(2)=−1f^{\prime}(2)=-1f′(2)=−1, and f(3)=1f(3)=1f(3)=1.

Problem 30539

Find the function f(x)f(x)f(x) given f′′(x)=2f^{\prime \prime}(x)=2f′′(x)=2, f′(1)=4f^{\prime}(1)=4f′(1)=4, and f(2)=−2f(2)=-2f(2)=−2.

Problem 30540

Problem 30541

Find the function f(x)f(x)f(x) given f′′(x)=1x3/2f^{\prime \prime}(x)=\frac{1}{x^{3/2}}f′′(x)=x3/21​, f′(4)=2f^{\prime}(4)=2f′(4)=2, and f(0)=1f(0)=1f(0)=1.

Problem 30542

Find the slope of secant lines for f(x)=(x−1)1/3f(x)=(x-1)^{1 / 3}f(x)=(x−1)1/3 between (1,0)(1,0)(1,0) and (x,f(x))(x,f(x))(x,f(x)) at x=1.5,1.1,1.01,1.0001x=1.5,1.1,1.01,1.0001x=1.5,1.1,1.01,1.0001.

Problem 30543

Find the derivative of the function $h(z) = 27z - z^3$ .

Find the extreme values of the function $y=x^{3}-12 x+2$ and identify where they occur.

Find the tangent line equation for the curve $f(x)=x^{2}+11 x-15$ at $x=1$ .

Find the tangent line equation for the function $f(x)=2 x^{2}-3 x+6$ at the point where $x=2$ .

Determine where the function $h(z) = 27z - z^3$ is increasing or decreasing on the intervals given.

Find $c$ such that $\frac{f(-1)-f(-2)}{-1-(-2)}=f(c)$ for $f(x)=x^{2}+5x+3$ in $[-2,-1]$ .

Find $c$ such that $\frac{f(8)-f(4)}{8-4}=f^{\prime}(c)$ for $f(x)=\ln (x-3)$ on $[4,8]$ . Round to the nearest thousandth.

Find the half-life of a drug if the initial level is $3,600 \mathrm{mg} / \mathrm{L}$ and after one day it is $1,160 \mathrm{mg} / \mathrm{L}$ .

Find the intervals where the function $f(x)=x^{3}+3 x^{2}-x-24$ is concave up or down.

Calculate the average rate of change of $f(x)=3 x^{2}-1$ over the interval $[-4,4]$ .

Water is wetting a circular area expanding at $5 \text{ mm}^2/\text{s}$ . Find the radius expansion rate at $155 \text{ mm}$ .

Find the tangent line equation at the point (0,1) for the curve $y^{4}+x^{3}=y^{2}+12 x$ .

Find the derivative of $f(x)=\frac{9 e^{x}}{2 e^{x}+1}$ .

Find the derivative of $y$ where $y=3 \sin^{-1}(5 x^{4})$ .

Find values of $c$ for $f(x)=x+\frac{45}{x}$ in $[5,9]$ that satisfy $\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$ .

Determine the intervals where the function $G(x)=\frac{1}{x^{2}}+7$ is increasing or decreasing.

Find critical points and the max/min values of $f(x)=x^{3}-12x-3$ on the interval $I=[-3,5]$ .

Find critical points and determine the max and min of $f(x)=x^{3}-12 x-3$ on the interval $I=[-3,5]$ .

Find the derivative of $y$ with respect to $x$ , $t$ , or $\theta$ : $y=\ln(6x)$ .

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

Determine where the function $G(x)=\frac{1}{4} x^{4}-x^{3}+15$ is concave up, concave down, and find inflection points.

Find the derivative of $y = x^{5} \ln x - \frac{1}{3} x^{3}$ with respect to $x$ .

Find the limit as $x$ approaches 2 from the right for the expression $3x + 1$ .

Find the slope of secant lines for $f(x)=(x-1)^{1/3}$ between $(1,0)$ and $(x,f(x))$ at $x=0.5,0.9,0.99,0.9999$ .

Find $f^{\prime \prime}(x)$ using the chain rule for $f^{\prime \prime}(x)=\frac{d}{d x}\left(\frac{2 x}{3\left(x^{2}-1\right)^{2 / 3}}\right)$ .

An object's height is $s(t)=125-5t^{2}$ . Find when it hits the ground, average velocity, and velocity at $t=4$ .

Find the limit as $x$ approaches 3 from the right: $\lim _{x \rightarrow 3^{+}}\left(\frac{x+2}{x-1}\right)$ .

Find the limit as $x$ approaches -1 from the left for the expression $1 - 5x$ .

Find the average rate of change of $f(x)=\frac{6}{3 x-7}$ on $[-3,0]$ . What is the average rate of change?

Find the midpoint Riemann sum for $\int_{2}^{3.5} x^{3} dx$ using 6 equal subintervals. Round to the nearest thousandth.

Find the function $f(x)$ given $f^{\prime \prime}(x)=2 x$ , $f^{\prime}(2)=-1$ , and $f(3)=1$ .

Find the function $f(x)$ given $f^{\prime \prime}(x)=2$ , $f^{\prime}(1)=4$ , and $f(2)=-2$ .

Find the function $f(x)$ given $f^{\prime \prime}(x)=\frac{1}{x^{3/2}}$ , $f^{\prime}(4)=2$ , and $f(0)=1$ .

Find the slope of secant lines for $f(x)=(x-1)^{1 / 3}$ between $(1,0)$ and $(x,f(x))$ at $x=1.5,1.1,1.01,1.0001$ .

Find the limits: $\lim_{x \to 4^-} f(x)$ , $\lim_{x \to 4^+} f(x)$ , and $\lim_{x \to 4} f(x)$ for the piecewise function $f(x)$ .

Show that the derivative of the loss function $E(\mathbf{w})$ satisfies $\frac{\partial y_{k}}{\partial a_{j}}=y_{k}(\delta_{kj}-y_{j})$ for softmax output units.

What do the secant lines indicate about the tangent line of the function $f(x)=(x-1)^{1/3}$ at $x=1$ ?

Find the limit: $\lim _{h \rightarrow 1} \frac{h^{2}+3 h-4}{h-1}$ . Options: A. 0, B. -3, C. 5, D. undefined.

Find the limit as $h$ approaches 0 of $\frac{f(x+h)-f(x)}{h} = \frac{2(x+h)\sqrt{2(x+h)+3} - 2x\sqrt{2x+3}}{h}$ .

Evaluate and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=7 x^{4}+x-8$ without remaining division by $h$ .

Find the radius $r$ and height $h$ of a cylindrical can using $1536 \pi$ in² of metal for maximum volume.

Calculate the account value after 11 years for an investment of \ $3950 at a 9.9% annual rate, using$ V=P e^{r t}$.

Evaluate the integral: $\int \frac{\sqrt{x^{3}}-\sqrt[3]{x}}{3 \sqrt[4]{x}} d x$

Find the integral: $\int \frac{\sqrt{x^{3}}-\sqrt[3]{x}}{3 \sqrt[4]{x}} d x$

Evaluate the function $f(x)=-x^{3}+5 x^{2}+2 x+5$ for: a. $f(x+h)$ b. $\frac{f(x+h)-f(x)}{h}$ (no $h$ in the answer).

Find the acceleration due to gravity at a distance of $1.83 \times 10^{8} \mathrm{~m}$ above Earth's surface. Units: $\mathrm{m} / \mathrm{s}^{\wedge} 2$ .

What is the robin's displacement $\Delta x$ from $t=0$ s to $t=1.5$ s based on its velocity graph? Answer with two significant digits.

When does a 15m falling object, with $s(t)=15-6.729 t^{2}$ , hit the ground? Round $t$ to 3 decimal places.

Find $y^{\prime}(e)$ for the function $y=e^{x-e}+\ln x-\frac{x}{e}$ .

Find the average velocity of an object falling from 15 m with $s(t)=15-6.729 t^{2}$ after $t$ seconds (round to 3 decimals).

Find the derivative of $F(t) = \ln(\sinh(3t))$ . What is $F'(t)$ ?

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

Find the derivative of $g(x)=\sinh ^{2}(x)$ . What is $g^{\prime}(x)=?$

Find the velocity of an object falling from 15 meters, given $s(t)=15-6.729 t^{2}$ , when it hits the ground.

Find the limit: $\lim _{x \rightarrow \infty} x \cdot \sin \left(\frac{\pi}{x}\right)$ .

Find $a$ if $\lim _{x \rightarrow-\infty} \frac{\sqrt{a x^{2}+b x+c}}{d-a x}=2$ .

Find the derivative $\frac{d y}{d x}$ for the equation $y=\log _{x} e$ .

Find the average rate of change of $f(x)=\left(\frac{1}{2}\right)^{x}$ from $x=-2$ to $x=2$ .

Evaluate the integral $\int_{-4}^{1} \sqrt{2-x} \, \mathrm{d} x$ and round to 3 significant figures.

Find the function $f(x)$ if $f^{\prime}(x)=4-6 x$ and $f(1)=0$ .

Find the derivative of $\sqrt{\frac{x-3}{2x+5}}$ using quotient and chain rule, showing all steps.