Calculus

Problem 1501

Estimate the average rate of change from $x=1$ to $x=5$ using $y=2$ and $y=3$ . Round to one decimal place.

See Solution

Problem 1502

Find the derivative $d y / d x$ using implicit differentiation for the equation $8 \cos (9 x) \sin (7 y) = 9$ .

See Solution

Problem 1503

Calculate the following integrals: (a) $\int x^{6} dx$ (b) $\int(6 x^{2}-2) dx$ (c) $\int\left(\frac{3}{2 x^{4}}+\frac{1}{\sqrt[3]{x}}+2\right) dx$ (d) $\int \frac{3+\sqrt{x}}{x^{2}} dx$ (e) $\int(3 x-2)^{7} dx$ (f) $\int \frac{1}{(5 x+4)^{4}} dx$ (g) $\int 4 x^{3}(x^{4}-3)^{5} dx$ (h) $\int \frac{1}{2 x+3} dx$ (i) $\int \frac{x^{2}}{5+x^{3}} dx$ (j) $\int e^{3 x+1} dx$ (k) $\int \frac{1-e^{x}}{e^{x}} dx$ (l) $\int x(e^{x^{2}}-2) dx$

See Solution

Problem 1504

Find the integral of $f(x)=\sin 2x$ . Options: $(-1/2)\cos 2x+C$ , $(1/2)\sin 2x+C$ , etc.

See Solution

Problem 1505

Find the secant line slope for $f(x)=\sqrt[3]{x}-11$ at $x_1=750$ and $x_2=6$ .

See Solution

Problem 1506

Find the limit: $\lim _{x \rightarrow 0}(\cot (6 x) \cdot \sin (2 x))$ .

See Solution

Problem 1507

Evaluate these integrals: (a) $\int_{0}^{3}(2-x) x^{2} d x$ , (b) $\int_{3}^{8} \sqrt{1+x} d x$ , (c) $\int_{0}^{1} x(x+2)^{5} d x$ , (d) $\int_{0}^{4}(6 x+1)(3 x^{2}+x) d x$ .

See Solution

Problem 1508

Find the left-hand and right-hand limits of $f(x) = \sin(x)$ as $x$ approaches 1, and the limit as $x$ approaches -1.

See Solution

Problem 1509

Find the limit: $\lim _{x \rightarrow 7} \frac{x^{3}-343}{x-7}$ .

See Solution

Problem 1510

Calculate the expected gradients for a neural network output $y=\sigma\left(\delta_{0} w_{0} x_{0}+\delta_{1} w_{1} x_{1}\right)$ with $\delta_{0}, \delta_{1} \sim \operatorname{Bernoulli}(0.5)$ :
1. $\mathbb{E}\left(\frac{\partial y}{\partial w_{0}}\right)$
2. $\mathbb{E}\left(\frac{\partial y}{\partial x_{0}}\right)$
3. $\mathbb{E}\left(\frac{\partial y}{\partial w_{1}}\right)$
4. $\mathbb{E}\left(\frac{\partial y}{\partial x_{1}}\right)$

See Solution

Problem 1511

Find the gradient of the output $f(\mathbf{x})$ with respect to $\mathbf{W}^{(2)}$ for the neural network defined as $f(\mathbf{x})=\sigma\left(\sigma\left(\mathbf{x} \cdot \mathbf{W}^{(1)}\right) \cdot \mathbf{W}^{(2)}\right)$ . Optional: Calculate the gradient with respect to $\mathbf{W}^{(1)}$ .

See Solution

Problem 1512

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

See Solution

Problem 1513

Find the limit: $\lim _{x \rightarrow \infty} \frac{(5-x^{2})(7 x^{2}-2)}{-8 x^{3}(x^{2}-10)}$ . State DNE if infinite.

See Solution

Problem 1514

Graph a function that is continuous but not differentiable at $x=3$ .

See Solution

Problem 1515

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}$ . State if it does not exist (DNE).

See Solution

Problem 1516

Find the limit: $\lim _{x \rightarrow \infty} \frac{-24 x^{3}-9 x^{2}}{2 x^{3}-x^{2}-2 x+1}$ . State if it DNE.

See Solution

Problem 1517

Find the limit: $\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}$ . State DNE if infinite.

See Solution

Problem 1518

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}$ . State DNE if infinite.

See Solution

Problem 1519

Find the limit: $\lim _{x \rightarrow \infty} \frac{40 x^{5}+15 x^{2}}{18 x^{3}-27 x^{5}+4-6 x^{2}}$ . State if it is infinite (DNE).

See Solution

Problem 1520

Find the limit: $\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}$ . State if it DNE.

See Solution

Problem 1521

Find the limit: $\lim _{x \rightarrow \infty} \frac{(7+5 x^{2})(4 x^{3}-7)}{(2 x^{3}+9)(1+9 x^{2})}$ . State if it DNE.

See Solution

Problem 1522

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}$ . State DNE if infinite.

See Solution

Problem 1523

Find the limit: $\lim _{x \rightarrow \infty} \frac{36+36 x+5 x^{2}}{90 x^{6}-29 x^{3}+2}$ . State if it DNE.

See Solution

Problem 1524

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}$ . State DNE if infinite.

See Solution

Problem 1525

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}$ . State if it DNE.

See Solution

Problem 1526

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}$ . State if it DNE.

See Solution

Problem 1527

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . If infinite, state DNE.

See Solution

Problem 1528

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . State if it DNE.

See Solution

Problem 1529

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . State DNE if infinite.

See Solution

Problem 1530

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-x^{33}-9 x^{20}}{x^{33}-5 \log _{5} x}$ . What does it equal?

See Solution

Problem 1531

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-6 x^{91}}{x^{19}+x^{91}}$ . What does it equal?

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Problem 1532

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-7 x^{46}}{\ln x 6 e^{x}}$ . What does it equal?

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Problem 1533

Find the horizontal asymptotes of the function $f(x)=1+2 e^{\frac{1}{x}}$ .

See Solution

Problem 1534

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}$ .

See Solution

Problem 1535

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}$ .

See Solution

Problem 1536

Evaluate the limit: $\lim_{x \to 0} \frac{-3x + 1 - \cos(x)}{2x}$

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Problem 1537

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{3 \sin ^{2}(x)}{4 x}$ .

See Solution

Problem 1538

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{-4 \sin (x)}{7 x}$ .

See Solution

Problem 1539

Find the horizontal asymptotes of the function $f(x)=1+2 e^{\frac{1}{x}}$ .

See Solution

Problem 1540

Find the horizontal asymptotes of the function $f(x)=\frac{3 x^{4}-4 x^{5}}{5 x^{3}-x^{4}}$ .

See Solution

Problem 1541

Find the horizontal asymptotes for the function $f(x)=2 e^{\frac{1}{x}}$ .

See Solution

Problem 1542

Hàm số $y=f(x)$ có đạo hàm $f^{\prime}(x)=x(x-1)(x-2)$ . Số khẳng định đúng về $f(x)$ là bao nhiêu? A. 4. B. 3. C. 2. D. 1.

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Problem 1543

Find the derivative $f^{\prime}(x)$ of $f(x)=x^{3}+4 \sqrt{x}+\frac{1}{x}+2$ and the tangent line at $x=1$ .

See Solution

Problem 1544

Find the $x$ -coordinates where the tangent of $f(x)=\frac{1}{3} x^{3}-5 x^{2}+1$ has slope $m=-16$ .

See Solution

Problem 1545

Find values of $A$ and $B$ for the function $f(x)=\left\{\begin{array}{ll}A x^{3}, & x<2 \\ B x^{2}+8, & x \geq 2\end{array}\right.$ to be differentiable everywhere.

See Solution

Problem 1546

Find the derivatives: 1) $y=5 t^{3}+t^{2}+4 t-1$ , find $\frac{d y}{d t}$ . 2) $y=2 u^{4}+4 u^{3}+4$ , find $\frac{d^{2} y}{d u^{2}}$ . 3) $y=2 \sin (\theta)+5 \cos (\theta)$ , find $\frac{d^{5} y}{d \theta^{5}}$ .

See Solution

Problem 1547

Find the derivative of the function: $\frac{d}{d x} \sqrt[4]{\frac{x^{3}+8}{x^{3}-8}}$ .

See Solution

Problem 1548

For the function $F(x)=-x^{4}+20 x^{2}+125$ , find if $F$ is even/odd, a second local max, and area under $F$ from $x=-5$ to $0$ .

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Problem 1549

How long (in yr) will it take for 1.00 g of Strontium-90 to decay to 0.200 g, given a half-life of 28.1 yr?

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Problem 1550

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{\sin \left(x^{2}+1\right)}{\sqrt{x}(5-x)}$

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Problem 1551

Find the value(s) of $x$ where the function $f(x)$ is discontinuous, defined as: $f(x) = \begin{cases} x+1 & \text{if } x<0 \\ 2 \cos (\pi x) & \text{if } 0 \leq x \leq 2 \\ 6-x^{2} & \text{if } x>2 \end{cases}$

See Solution

Problem 1552

Evaluate $\lim _{x \rightarrow 0} f(x)$ given $2-\cos x \leq f(x) \leq x^{2}+1$ .

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Problem 1553

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (6 x)}{\sin (2 x)}$ .

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Problem 1554

Solve the initial value problem $y' = 9ty^2$ , $y(0) = y_0$ , and find how the solution interval depends on $y_0$ .

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Problem 1555

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{12}{x}$ and evaluate $f^{\prime}(-2), f^{\prime}(0), f^{\prime}(5)$ .

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Problem 1556

Find the derivative $f^{\prime}(x)$ for $f(x)=\frac{48}{x}$ and calculate $f^{\prime}(-4)$ , $f^{\prime}(0)$ , $f^{\prime}(3)$ .

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Problem 1557

Find the derivative $g^{\prime}(x)$ for $g(x)=\sqrt{7 x}$ , then calculate $g^{\prime}(-3), g^{\prime}(0), g^{\prime}(4)$ .

See Solution

Problem 1558

Solve the initial value problem $y^{\prime}+7 y=g(t)$ with $y(0)=0$ , where $g(t)=1$ for $0 \leq t \leq 1$ and $g(t)=0$ for $t>1$ .

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Problem 1559

Find the derivative $g^{\prime}(x)$ of $g(x)=\sqrt{15 x}$ , then compute $g^{\prime}(-3)$ , $g^{\prime}(0)$ , and $g^{\prime}(3)$ .

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Problem 1560

Find the derivative $f^{\prime}(x)$ of $f(x)=4x^{3}+10$ using the limit definition, then calculate $f^{\prime}(-1)$ , $f^{\prime}(0)$ , and $f^{\prime}(4)$ .

See Solution

Problem 1561

Find the secant line for $y=f(x)=x^{2}+x$ at $x=3$ and $x=7$ , and the tangent line at $x=3$ .

See Solution

Problem 1562

Find the slope of the secant line for $y=f(x)=x^{2}+x$ at $x=-3$ and $x=-1$ . Also, find the tangent line at $x=-3$ .

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Problem 1563

Find the secant line between $x=3$ and $x=7$ for $y=f(x)=x^{2}+x$ , and the tangent line at $x=3$ .

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Problem 1564

Find the secant line between $x=3$ and $x=7$ for $y=f(x)=x^{2}+x$ and the tangent line at $x=3$ .

See Solution

Problem 1565

Find the secant line for $y=f(x)=x^{2}+x$ at $x=3$ and $x=7$ , and the tangent line at $x=3$ .

See Solution

Problem 1566

Find the secant line between $x=-3$ and $x=-1$ for $y=f(x)=x^{2}+x$ and the tangent line at $x=-3$ .

See Solution

Problem 1567

Find the slope of the tangent line for $f(x)=\frac{5}{x}$ at $x=4$ by calculating the instantaneous rate of change.

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Problem 1568

Find the slope of the tangent line for $f(x)=\frac{5}{x}$ at $x=4$ . The slope is $-\frac{5}{16}$ . What is the tangent line equation?

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Problem 1569

Find the $x$ -coordinates where the slope of the tangent to $f(x)=\frac{1}{3} x^{3}-5 x^{2}+1$ is $m=-16$ .

See Solution

Problem 1570

Find the secant line for $f(x)=\frac{5}{x}$ at $x=4$ and $x=5$ , and the tangent line at $x=4$ .

See Solution

Problem 1571

Find $\delta$ such that if $|x-1|<\delta$ , then $|f(x)-1|<0.2$ , given $f(1)=1$ , $f(1.1)=0.8$ , $f(1.2)=1.2$ .

See Solution

Problem 1572

Find $\delta$ so that if $0<|x-3|<\delta$ , then $|f(x)-2|<0.5$ for the function $f$ at points $(2.6,1.5)$ and $(3.8,2.5)$ .

See Solution

Problem 1573

Find $\delta$ such that if $|x-4|<\delta$ , then $|\sqrt{x}-2|<0.4$ for $f(x)=\sqrt{x}$ .

See Solution

Problem 1574

Find $\delta$ so that if $|x-1|<\delta$ , then $|f(x)-1|<0.2$ using points (.7,1.2) and (1.1,.8).

See Solution

Problem 1575

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (4 x)}{2 x}=$

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Problem 1576

Find $\delta$ so that if $0<|x-3|<\delta$ , then $|f(x)-2|<0.5$ using the graph of $f$ .

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Problem 1577

Identify which functions are continuous for all real numbers: I. $f(x)=x^{1 / 3}$ II. $g(x)=\sec x$ III. $h(x)=e^{-x}$ (A) I only (B) I and II only (C) I and III only (D) I, II, and III

See Solution

Problem 1578

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos x}{3 \sin ^{2} x}$ .

See Solution

Problem 1579

Find the derivative of $f(x)=-x^{7}-x^{6}+\frac{3}{2}$ at $x=-3$ . What is $f^{\prime}(-3)=?$

See Solution

Problem 1580

Find the slope of the tangent line to $f(x)=x^{3}-4 x+2$ at $x=2$ .

See Solution

Problem 1581

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (2 x)}{3 x}=$ (A) $\frac{1}{3}$ (B) $\frac{1}{2}$ (C) $\frac{2}{3}$ (D) 2

See Solution

Problem 1582

Find the tangent line equation for $f(x)=2 x^{3}+3 x-3$ at the point where $x=1$ .

See Solution

Problem 1583

Find the tangent line equation for $f(x)=-x^{2}+4x+2$ at $x=-1$ .

See Solution

Problem 1584

Find the derivatives: 1) $y=5 t^{3}+t^{2}+4 t-1$ , find $\frac{d y}{d t}$ ; 2) $y=2 u^{4}+4 u^{3}+4$ , find $\frac{d^{2} y}{d u^{2}}$ ; 3) $y=2 \sin (\theta)+5 \cos (\theta)$ , find $\frac{d^{5} y}{d \theta^{5}}$ .

See Solution

Problem 1585

Find the limit: $\lim _{x \rightarrow \infty} \frac{8 x^{2}-5}{5 x^{2}+x-3}$ .

See Solution

Problem 1586

Find the limit: $\lim _{x \rightarrow-\infty} \frac{6 x^{5}-x}{x^{4}+3}$ .

See Solution

Problem 1587

Find the derivative $f^{\prime}\left(\frac{\pi}{4}\right)$ for the function $f(x)=\sin x(2+\cos x)$ .

See Solution

Problem 1588

Find the derivative $\frac{d y}{d x}$ for $y=\sin x-\cos 3 x$ .

See Solution

Problem 1589

Find $\frac{d y}{d x}$ for $y=\sin x-\cos 3 x$ at $x=45^{\circ}$ .

See Solution

Problem 1590

Find $\left.\frac{d y}{d x}\right|_{45^{\circ}}$ for $y=\sin x-\cos 3 x$ .

See Solution

Problem 1591

Cari turunan $f^{\prime}(x)$ dari fungsi $f(x)=\sin ^{2} x+\cos 3 x$ .

See Solution

Problem 1592

Find the derivative of $y=\sin 3x - \cos 3x$ at $x=45^{\circ}$ .

See Solution

Problem 1593

Find the derivative of the function $f(x)=\sqrt{x} \cos^{3} x$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 1594

Find the derivative of the function $y=\frac{\sin x+\cos x}{x}$ .

See Solution

Problem 1595

Find the derivative $f^{\prime}(\pi)$ for the function $f(x)=\frac{1}{1+\cos x}$ .

See Solution

Problem 1596

Determine the slope, $m$ , of the tangent line to the curve $y=7+5 x^{2}-2 x^{3}$ at $x=a$ .

See Solution

Problem 1597

Compute $\Delta y / \Delta x$ for $y=4x-7$ over $[2,6]$ and find the instantaneous rate of change at $x=2$ .

See Solution

Problem 1598

Find the velocity of a particle with displacement $s=\frac{2}{t^{2}}$ at $t=a, t=1, t=2, t=3$ (in $\mathrm{m/s}$ ).

See Solution

Problem 1599

Find the critical points of $f(x, y)=x^{2}+2xy+2y^{2}-8x+2y$ and classify each as max, min, or saddle.

See Solution

Problem 1600

Find critical points of $t(x, y)=x^{3}-12xy+y^{3}$ and classify as max, min, or saddle.

See Solution

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Calculus

Problem 1501

Estimate the average rate of change from x=1x=1x=1 to x=5x=5x=5 using y=2y=2y=2 and y=3y=3y=3. Round to one decimal place.

Problem 1502

Find the derivative dy/dxd y / d xdy/dx using implicit differentiation for the equation 8cos⁡(9x)sin⁡(7y)=98 \cos (9 x) \sin (7 y) = 98cos(9x)sin(7y)=9.

Problem 1503

Problem 1504

Find the integral of f(x)=sin⁡2xf(x)=\sin 2xf(x)=sin2x. Options: (−1/2)cos⁡2x+C(-1/2)\cos 2x+C(−1/2)cos2x+C, (1/2)sin⁡2x+C(1/2)\sin 2x+C(1/2)sin2x+C, etc.

Problem 1505

Find the secant line slope for f(x)=x3−11f(x)=\sqrt[3]{x}-11f(x)=3x​−11 at x1=750x_1=750x1​=750 and x2=6x_2=6x2​=6.

Problem 1506

Find the limit: lim⁡x→0(cot⁡(6x)⋅sin⁡(2x))\lim _{x \rightarrow 0}(\cot (6 x) \cdot \sin (2 x))limx→0​(cot(6x)⋅sin(2x)).

Problem 1507

Evaluate these integrals: (a) ∫03(2−x)x2dx\int_{0}^{3}(2-x) x^{2} d x∫03​(2−x)x2dx, (b) ∫381+xdx\int_{3}^{8} \sqrt{1+x} d x∫38​1+x​dx, (c) ∫01x(x+2)5dx\int_{0}^{1} x(x+2)^{5} d x∫01​x(x+2)5dx, (d) ∫04(6x+1)(3x2+x)dx\int_{0}^{4}(6 x+1)(3 x^{2}+x) d x∫04​(6x+1)(3x2+x)dx.

Problem 1508

Find the left-hand and right-hand limits of f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x) as xxx approaches 1, and the limit as xxx approaches -1.

Problem 1509

Find the limit: lim⁡x→7x3−343x−7\lim _{x \rightarrow 7} \frac{x^{3}-343}{x-7}limx→7​x−7x3−343​.

Problem 1510

Problem 1511

Problem 1512

Find the limit of f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ as h→0h \to 0h→0 for f(x)=2x+3f(x) = 2x + 3f(x)=2x+3.

Problem 1513

Find the limit: lim⁡x→∞(5−x2)(7x2−2)−8x3(x2−10)\lim _{x \rightarrow \infty} \frac{(5-x^{2})(7 x^{2}-2)}{-8 x^{3}(x^{2}-10)}x→∞lim​−8x3(x2−10)(5−x2)(7x2−2)​. State DNE if infinite.

Problem 1514

Graph a function that is continuous but not differentiable at x=3x=3x=3.

Problem 1515

Find the limit as xxx approaches infinity: lim⁡x→∞(1−7x3)(x−8)(3x2−1)(2x2+3)\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}x→∞lim​(3x2−1)(2x2+3)(1−7x3)(x−8)​. State if it does not exist (DNE).

Problem 1516

Find the limit: lim⁡x→∞−24x3−9x22x3−x2−2x+1\lim _{x \rightarrow \infty} \frac{-24 x^{3}-9 x^{2}}{2 x^{3}-x^{2}-2 x+1}x→∞lim​2x3−x2−2x+1−24x3−9x2​. State if it DNE.

Problem 1517

Find the limit: lim⁡x→∞(1−7x3)(x−8)(3x2−1)(2x2+3)\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}x→∞lim​(3x2−1)(2x2+3)(1−7x3)(x−8)​. State DNE if infinite.

Problem 1518

Find the limit as xxx approaches infinity: lim⁡x→∞(1+10x3)(4+x2)(10−x3)(x2−3)\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}x→∞lim​(10−x3)(x2−3)(1+10x3)(4+x2)​. State DNE if infinite.

Problem 1519

Find the limit: lim⁡x→∞40x5+15x218x3−27x5+4−6x2\lim _{x \rightarrow \infty} \frac{40 x^{5}+15 x^{2}}{18 x^{3}-27 x^{5}+4-6 x^{2}}x→∞lim​18x3−27x5+4−6x240x5+15x2​. State if it is infinite (DNE).

Problem 1520

Find the limit: lim⁡x→∞(1+10x3)(4+x2)(10−x3)(x2−3)\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}x→∞lim​(10−x3)(x2−3)(1+10x3)(4+x2)​. State if it DNE.

Problem 1521

Find the limit: lim⁡x→∞(7+5x2)(4x3−7)(2x3+9)(1+9x2)\lim _{x \rightarrow \infty} \frac{(7+5 x^{2})(4 x^{3}-7)}{(2 x^{3}+9)(1+9 x^{2})}x→∞lim​(2x3+9)(1+9x2)(7+5x2)(4x3−7)​. State if it DNE.

Problem 1522

Find the limit: lim⁡x→∞−2x2+x3−33x10x2+6\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}x→∞lim​10x2+6−2x2+x3−33x​​. State DNE if infinite.

Problem 1523

Find the limit: lim⁡x→∞36+36x+5x290x6−29x3+2\lim _{x \rightarrow \infty} \frac{36+36 x+5 x^{2}}{90 x^{6}-29 x^{3}+2}x→∞lim​90x6−29x3+236+36x+5x2​. State if it DNE.

Problem 1524

Find the limit: lim⁡x→∞42x2+16x105x+4x4+4x2\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}x→∞lim​5x+4x4+4x242x2+16x10​​. State DNE if infinite.

Problem 1525

Find the limit: lim⁡x→∞−2x2+x3−33x10x2+6\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}x→∞lim​10x2+6−2x2+x3−33x​​. State if it DNE.

Problem 1526

Find the limit as xxx approaches infinity: lim⁡x→∞42x2+16x105x+4x4+4x2\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}x→∞lim​5x+4x4+4x242x2+16x10​​. State if it DNE.

Problem 1527

Find the limit as xxx approaches infinity: lim⁡x→∞−5x4+33x5−64x7310x+8+9x3\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}x→∞lim​10x+8+9x33−5x4+33x5−64x7​​. If infinite, state DNE.

Problem 1528

Find the limit: lim⁡x→∞−5x4+33x5−64x7310x+8+9x3\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}x→∞lim​10x+8+9x33−5x4+33x5−64x7​​. State if it DNE.

Problem 1529

Find the limit: lim⁡x→∞−5x4+33x5−64x7310x+8+9x3\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}x→∞lim​10x+8+9x33−5x4+33x5−64x7​​. State DNE if infinite.

Problem 1530

Determine the limit: lim⁡x→∞−x33−9x20x33−5log⁡5x\lim _{x \rightarrow \infty} \frac{-x^{33}-9 x^{20}}{x^{33}-5 \log _{5} x}x→∞lim​x33−5log5​x−x33−9x20​. What does it equal?

Problem 1531

Determine the limit: lim⁡x→∞−6x91x19+x91\lim _{x \rightarrow \infty} \frac{-6 x^{91}}{x^{19}+x^{91}}x→∞lim​x19+x91−6x91​. What does it equal?

Problem 1532

Determine the limit: lim⁡x→∞−7x46ln⁡x6ex\lim _{x \rightarrow \infty} \frac{-7 x^{46}}{\ln x 6 e^{x}}x→∞lim​lnx6ex−7x46​. What does it equal?

Problem 1533

Find the horizontal asymptotes of the function f(x)=1+2e1xf(x)=1+2 e^{\frac{1}{x}}f(x)=1+2ex1​.

Problem 1534

Evaluate the limit: lim⁡x→07sin⁡(3x)3x\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}limx→0​3x7sin(3x)​.

Problem 1535

Find the limit as xxx approaches 0: lim⁡x→07sin⁡(3x)3x\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}limx→0​3x7sin(3x)​.

Problem 1536

Evaluate the limit: lim⁡x→0−3x+1−cos⁡(x)2x\lim_{x \to 0} \frac{-3x + 1 - \cos(x)}{2x}limx→0​2x−3x+1−cos(x)​

Problem 1537

Evaluate the limit: lim⁡x→03sin⁡2(x)4x\lim _{x \rightarrow 0} \frac{3 \sin ^{2}(x)}{4 x}limx→0​4x3sin2(x)​.

Problem 1538

Evaluate the limit: lim⁡x→0−4sin⁡(x)7x\lim _{x \rightarrow 0} \frac{-4 \sin (x)}{7 x}limx→0​7x−4sin(x)​.

Problem 1539

Find the horizontal asymptotes of the function f(x)=1+2e1xf(x)=1+2 e^{\frac{1}{x}}f(x)=1+2ex1​.

Problem 1540

Find the horizontal asymptotes of the function f(x)=3x4−4x55x3−x4f(x)=\frac{3 x^{4}-4 x^{5}}{5 x^{3}-x^{4}}f(x)=5x3−x43x4−4x5​.

Problem 1541

Find the horizontal asymptotes for the function f(x)=2e1xf(x)=2 e^{\frac{1}{x}}f(x)=2ex1​.

Estimate the average rate of change from $x=1$ to $x=5$ using $y=2$ and $y=3$ . Round to one decimal place.

Find the derivative $d y / d x$ using implicit differentiation for the equation $8 \cos (9 x) \sin (7 y) = 9$ .

Find the integral of $f(x)=\sin 2x$ . Options: $(-1/2)\cos 2x+C$ , $(1/2)\sin 2x+C$ , etc.

Find the secant line slope for $f(x)=\sqrt[3]{x}-11$ at $x_1=750$ and $x_2=6$ .

Find the limit: $\lim _{x \rightarrow 0}(\cot (6 x) \cdot \sin (2 x))$ .

Evaluate these integrals: (a) $\int_{0}^{3}(2-x) x^{2} d x$ , (b) $\int_{3}^{8} \sqrt{1+x} d x$ , (c) $\int_{0}^{1} x(x+2)^{5} d x$ , (d) $\int_{0}^{4}(6 x+1)(3 x^{2}+x) d x$ .

Find the left-hand and right-hand limits of $f(x) = \sin(x)$ as $x$ approaches 1, and the limit as $x$ approaches -1.

Find the limit: $\lim _{x \rightarrow 7} \frac{x^{3}-343}{x-7}$ .

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

Find the limit: $\lim _{x \rightarrow \infty} \frac{(5-x^{2})(7 x^{2}-2)}{-8 x^{3}(x^{2}-10)}$ . State DNE if infinite.

Graph a function that is continuous but not differentiable at $x=3$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}$ . State if it does not exist (DNE).

Find the limit: $\lim _{x \rightarrow \infty} \frac{-24 x^{3}-9 x^{2}}{2 x^{3}-x^{2}-2 x+1}$ . State if it DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}$ . State DNE if infinite.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}$ . State DNE if infinite.

Find the limit: $\lim _{x \rightarrow \infty} \frac{40 x^{5}+15 x^{2}}{18 x^{3}-27 x^{5}+4-6 x^{2}}$ . State if it is infinite (DNE).

Find the limit: $\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}$ . State if it DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{(7+5 x^{2})(4 x^{3}-7)}{(2 x^{3}+9)(1+9 x^{2})}$ . State if it DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}$ . State DNE if infinite.

Find the limit: $\lim _{x \rightarrow \infty} \frac{36+36 x+5 x^{2}}{90 x^{6}-29 x^{3}+2}$ . State if it DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}$ . State DNE if infinite.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}$ . State if it DNE.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}$ . State if it DNE.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . If infinite, state DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . State if it DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . State DNE if infinite.

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-x^{33}-9 x^{20}}{x^{33}-5 \log _{5} x}$ . What does it equal?

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-6 x^{91}}{x^{19}+x^{91}}$ . What does it equal?

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-7 x^{46}}{\ln x 6 e^{x}}$ . What does it equal?

Find the horizontal asymptotes of the function $f(x)=1+2 e^{\frac{1}{x}}$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}$ .

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}$ .

Evaluate the limit: $\lim_{x \to 0} \frac{-3x + 1 - \cos(x)}{2x}$

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{3 \sin ^{2}(x)}{4 x}$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{-4 \sin (x)}{7 x}$ .

Find the horizontal asymptotes of the function $f(x)=1+2 e^{\frac{1}{x}}$ .

Find the horizontal asymptotes of the function $f(x)=\frac{3 x^{4}-4 x^{5}}{5 x^{3}-x^{4}}$ .

Find the horizontal asymptotes for the function $f(x)=2 e^{\frac{1}{x}}$ .

Hàm số $y=f(x)$ có đạo hàm $f^{\prime}(x)=x(x-1)(x-2)$ . Số khẳng định đúng về $f(x)$ là bao nhiêu? A. 4. B. 3. C. 2. D. 1.

Find the derivative $f^{\prime}(x)$ of $f(x)=x^{3}+4 \sqrt{x}+\frac{1}{x}+2$ and the tangent line at $x=1$ .

Find the $x$ -coordinates where the tangent of $f(x)=\frac{1}{3} x^{3}-5 x^{2}+1$ has slope $m=-16$ .

Find values of $A$ and $B$ for the function $f(x)=\left\{\begin{array}{ll}A x^{3}, & x<2 \\ B x^{2}+8, & x \geq 2\end{array}\right.$ to be differentiable everywhere.

Find the derivative of the function: $\frac{d}{d x} \sqrt[4]{\frac{x^{3}+8}{x^{3}-8}}$ .

For the function $F(x)=-x^{4}+20 x^{2}+125$ , find if $F$ is even/odd, a second local max, and area under $F$ from $x=-5$ to $0$ .

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{\sin \left(x^{2}+1\right)}{\sqrt{x}(5-x)}$

Evaluate $\lim _{x \rightarrow 0} f(x)$ given $2-\cos x \leq f(x) \leq x^{2}+1$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (6 x)}{\sin (2 x)}$ .

Solve the initial value problem $y' = 9ty^2$ , $y(0) = y_0$ , and find how the solution interval depends on $y_0$ .

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{12}{x}$ and evaluate $f^{\prime}(-2), f^{\prime}(0), f^{\prime}(5)$ .

Find the derivative $f^{\prime}(x)$ for $f(x)=\frac{48}{x}$ and calculate $f^{\prime}(-4)$ , $f^{\prime}(0)$ , $f^{\prime}(3)$ .

Find the derivative $g^{\prime}(x)$ for $g(x)=\sqrt{7 x}$ , then calculate $g^{\prime}(-3), g^{\prime}(0), g^{\prime}(4)$ .

Solve the initial value problem $y^{\prime}+7 y=g(t)$ with $y(0)=0$ , where $g(t)=1$ for $0 \leq t \leq 1$ and $g(t)=0$ for $t>1$ .

Find the derivative $g^{\prime}(x)$ of $g(x)=\sqrt{15 x}$ , then compute $g^{\prime}(-3)$ , $g^{\prime}(0)$ , and $g^{\prime}(3)$ .

Find the derivative $f^{\prime}(x)$ of $f(x)=4x^{3}+10$ using the limit definition, then calculate $f^{\prime}(-1)$ , $f^{\prime}(0)$ , and $f^{\prime}(4)$ .

Find the secant line for $y=f(x)=x^{2}+x$ at $x=3$ and $x=7$ , and the tangent line at $x=3$ .

Find the slope of the secant line for $y=f(x)=x^{2}+x$ at $x=-3$ and $x=-1$ . Also, find the tangent line at $x=-3$ .

Find the secant line between $x=3$ and $x=7$ for $y=f(x)=x^{2}+x$ , and the tangent line at $x=3$ .

Find the secant line between $x=3$ and $x=7$ for $y=f(x)=x^{2}+x$ and the tangent line at $x=3$ .

Find the secant line for $y=f(x)=x^{2}+x$ at $x=3$ and $x=7$ , and the tangent line at $x=3$ .

Find the secant line between $x=-3$ and $x=-1$ for $y=f(x)=x^{2}+x$ and the tangent line at $x=-3$ .

Find the slope of the tangent line for $f(x)=\frac{5}{x}$ at $x=4$ by calculating the instantaneous rate of change.

Find the slope of the tangent line for $f(x)=\frac{5}{x}$ at $x=4$ . The slope is $-\frac{5}{16}$ . What is the tangent line equation?

Find the $x$ -coordinates where the slope of the tangent to $f(x)=\frac{1}{3} x^{3}-5 x^{2}+1$ is $m=-16$ .

Find the secant line for $f(x)=\frac{5}{x}$ at $x=4$ and $x=5$ , and the tangent line at $x=4$ .

Find $\delta$ such that if $|x-1|<\delta$ , then $|f(x)-1|<0.2$ , given $f(1)=1$ , $f(1.1)=0.8$ , $f(1.2)=1.2$ .

Find $\delta$ so that if $0<|x-3|<\delta$ , then $|f(x)-2|<0.5$ for the function $f$ at points $(2.6,1.5)$ and $(3.8,2.5)$ .

Find $\delta$ such that if $|x-4|<\delta$ , then $|\sqrt{x}-2|<0.4$ for $f(x)=\sqrt{x}$ .

Find $\delta$ so that if $|x-1|<\delta$ , then $|f(x)-1|<0.2$ using points (.7,1.2) and (1.1,.8).