Calculus

Problem 201

Differentiate the following expressions, leaving answers in simplified form with positive exponents: a) $g(x)=(x^2+4)^3$ (2 marks) b) $y=x^4(\sqrt{1+4x^2})$ (4 marks)

See Solution

Problem 202

Evaluate the integral $\int \frac{x^{3}+x^{15}}{\sqrt{4 x^{4}+x^{16}-1}} d x$ using a substitution, and find the constant of integration.

See Solution

Problem 203

Find the time(s) when a ball thrown from 5 ft with 23 ft/s upward velocity reaches 13 ft. $h=5+23t-16t^2$ , round to nearest hundredth.

See Solution

Problem 204

Tìm giá trị nhỏ nhất của hàm số $y=x+\sqrt{9-x^{2}}$ .

See Solution

Problem 205

Evaluate the indefinite integral $\int \frac{dx}{\sqrt{9x^2 - 4}}$ .

See Solution

Problem 206

Find the equation of the normal line to the function $f(x)$ at $x=1$ given $f(1)=2$ , $f'(1)=4$ , and $f'(2)=1$ .

See Solution

Problem 207

Find $z(x_0=1)$ and $z'(x_0=1)$ for $x^{-8} \cdot z^2 = 9$ . Round answers to 2 decimal places.

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Problem 208

Find the value of $y$ when $x=0$ in the equation $y=e^{x}+1$ .

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Problem 209

Find the slope of the oblique asymptote given by the function $f(x) = (ax^2 + bx + c) / (dx + e)$ .

See Solution

Problem 210

Find $s$ so $g(x) = s x^2 + 3x + 3$ has a minimum at $x = -1$ .

See Solution

Problem 211

Find the indicated quantities for $y=f(x)=8-x^{2}$ : (A) $\frac{f(-1)-f(-2)}{(-1)-(-2)}$ , (B) $\frac{f(-2+h)-f(-2)}{h}$ , (C) $\lim_{h\to 0} \frac{f(-2+h)-f(-2)}{h}$ .

See Solution

Problem 212

Find an exponential function with horizontal asymptote $y=-9$ and passing through points $(0,-8)$ and $(1,3)$ . $f(x)=a\cdot b^x$

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Problem 213

Estimate the area under $f(x)=\frac{1}{x+1}$ on $[3,5]$ using 8 right-endpoint rectangles. Repeat with left endpoints. $R_{n}$ and $L_{n}$ formulas.

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Problem 214

Graph the function $h(x) = -\frac{1}{x^2}$ . Find its domain, $x$ -intercepts, $y$ -intercept, and vertical asymptotes.
The domain of the function is $\mathbb{R} \setminus \{0\}$ . The $x$ -intercept(s) is/are $(0,0)$ . The $y$ -intercept is $(0,-1)$ . The function has one vertical asymptote at $x = 0$ .

See Solution

Problem 215

Find the 6th term in the Taylor series expansion of $f(x)=3e^{4x}$ around $x=4$ .

See Solution

Problem 216

Find the antiderivatives of $f(x)=-8 \sec^2 x$ and verify the solution by taking the derivative.

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Problem 217

Estimate the value of $y$ using the cubic model $y=10x^3-12x$ when $x=3$ .

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Problem 218

Find the domain of the function $f(x) = \log_6(-5x+4)$ .

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Problem 219

Solve the differential equation $4 e^{t} \cdot(e^{t}-1)=0$ for the value of $t$ .

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Problem 220

Evaluate $f \circ g \circ h(16)$ where $f(x) = x^2 + 9$ , $g(x) = x - 5$ , and $h(x) = \sqrt{x}$ .

See Solution

Problem 221

Find the particular solution form using the method of undetermined coefficients for $y'''+ 4y'' - 5y = xe^x + 11x$ . The form is $y_p(x) = A x e^x + B x$ .

See Solution

Problem 222

Find the inverse function $f^{-1}(x)$ of $f(x)=9+\sqrt{5x-3}$ .

See Solution

Problem 223

Select the correct statement about the radical function $f(x)=3 \sqrt[3]{-x}+2$ . A) Strictly increasing B) Strictly decreasing C) Increasing and decreasing D) Constant

See Solution

Problem 224

Calcul de l'intégrale de $f(x) = \sqrt{x^2 - 2x}$ sur $[2, 5]$ par la méthode de Simpson avec 6 sous-intervalles. Donnez l'approximation à 3 décimales près.

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Problem 225

Find a polynomial function $g$ where $\lim_{x\to-\infty} g(x) = -\infty$ . Which expression could define $g(x)$ ?

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Problem 226

Determine the end behavior of $f(x) = \log(x)$ as $x$ approaches $\infty$ .

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Problem 227

Find the function $y=f(u)$ and $u=g(x)$ , then compute $\frac{dy}{dx}$ as a function of $x$ , where $y=\cot(5x-6)$ .

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Problem 228

Find the equation of the normal line to the curve $y=2x^2-4x+1$ at the point $(2,1)$ in the form $y=mx+c$ .

See Solution

Problem 229

Find the simplified difference quotient $\frac{f(x+h)-f(x)}{h}$ and complete the table for $f(x)=x^2$ , $x=5$ , and various $h$ values.

See Solution

Problem 230

Find the limit of $\sqrt{x+4}$ as $x$ approaches 0.

See Solution

Problem 231

Find the relative maximum point of $f(x) = -x^3 + 3x^2 - 4$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 232

Find the domain, limits, and graph of the piecewise function $f(x) = \sqrt{25 - x^2}$ for $0 \leq x < 5$ , $f(x) = 5$ for $5 \leq x < 10$ , and $f(x) = 10$ for $x = 10$ .

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Problem 233

Find the slope of the secant line connecting $(-4, f(-4))$ and $(-2, f(-2))$ for $f(x)=-x^{2}+4 x+5$ on $[-4,-2]$ . Use the Mean Value Theorem to find all $c$ in $(-4,-2)$ such that $m=f^{\prime}(c)$ .
$m=\frac{f(-2)-f(-4)}{-2-(-4)}$ $c=\{\text{solutions}\}$

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Problem 234

Find the derivative of $y = \sin^n(x)\cos^n(x)$ for $n>2$ .

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Problem 235

Evaluate the indefinite integral of $12x^3 - 12x^5 + \sqrt{x}$ .

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Problem 236

Graph $f(x)=|4x+12|$ and find $\lim_{x\to-3}f(x)$ and $\lim_{x\to-5}f(x)$ . If necessary, state that the limit does not exist.

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Problem 237

Find the equation of the tangent line to $y=x^2-3$ at $x=4$ . A. $y=8x-19$ B. $y=8x-35$ C. $y=4x-19$ D. $y=8x-38$

See Solution

Problem 238

Evaluate the limit of $x^2 + \frac{2}{x+4}$ as $x$ approaches -4 from the left. Determine the limits of $f(x)$ and $g(x)$ as $x$ approaches $c$ , and evaluate the limits of their sum, product, and ratio.

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Problem 239

Find the value of $f(g(256))$ and $g(f(-4))$ for the functions $f(x)=2^{x}$ and $g(x)=\log_{4} x$ .

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Problem 240

Evaluate $\sqrt{61} \ln e^{\sqrt{61}}$ without a calculator and simplify the result.

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Problem 241

Evaluate the integral $\int \cos 3x \cos 3x dx$ and select the correct solution from the given options.

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Problem 242

Find the derivative of $3^x$ with respect to $x$ . Type $\ln(x)$ for the natural logarithm function.

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Problem 243

Calculate the integral $\int 2 \sqrt[3]{x} \, dx$ and express the result in simplest form.

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Problem 244

Calculate the integral of $-2 x^{-2} + 2 - 4 x$ and simplify the result.

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Problem 245

Find the limit of $\frac{\tan(2x)}{x\cos(x)}$ as $x$ approaches 0.

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Problem 246

Solve for $x$ in the equation $5 \ln x = -15$ . Select the correct choice: A. $x = e^{-3}$ , or B. The solution is not a real number.

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Problem 247

Find the critical points of the polynomial function $f(x) = x^4 + 4x^3 - 9$ .

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Problem 248

Find the area between the cubic curves $y=x^{3}-13x^{2}+30x$ and $y=-x^{3}+13x^{2}-30x$ .

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Problem 249

Sketch the graph of $f(x)=-e^{x-3}$ using transformations of $y=e^x$ . Find the domain, range, $y$ -intercept, and horizontal asymptote.

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Problem 250

Find the derivative of $\sin(g(x))$ evaluated at $x=8$ , given that $g(8)=3$ and $g'(8)=9$ . Round the answer to three decimal places.

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Problem 251

Find the range of $g(x) = -8^x$ . (Answer in interval notation)

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Problem 252

Differentiate the function $f(t) = \frac{\sec t}{-1 + \sec t}$ and find the derivative $f'(t)$ .

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Problem 253

Find the intervals where the function $f(x) = (9 - 6x)e^x$ is concave up and concave down.

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Problem 254

Find the derivative of the inverse function of $f(x) = 4x - 9$ at the point $(-13, -1)$ on the inverse function's graph.

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Problem 255

Find the expression that equals $(t \circ b \circ v)(x)$ given $b(x)=\sqrt{x}$ , $t(x)=2^{x}$ , and $v(x)=2|x|$ .
Answer options: $2^{\sqrt{2|x|}}$ $\sqrt{2\left|2^{x}\right|}$ $2\left|2^{\sqrt{x}}\right|$ $2\left|\sqrt{2^{x}}\right|$

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Problem 256

Estimate the errors in approximating the integral $\int_{2}^{4} \frac{1}{(x-1)^{2}} dx$ using the Trapezoidal Rule and Simpson's Rule with $n=4$ .

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Problem 257

Find the value of $y$ if $x = e^y$ .

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Problem 258

Find the domain of the inverse function $f^{-1}$ given that the domain of $f$ is $[5, \infty)$ and its range is $[-9, \infty)$ .
The domain of $f^{-1}$ is $[-9, \infty)$ .

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Problem 259

Find the derivative of $t(x)=0.08 e^{x} x^{2}$ evaluated at $x=5$ , rounded to the nearest tenth.

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Problem 260

Find the domain of the function $g(x) = \sqrt{8x+1}$ .

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Problem 261

Find the volume of the solid formed by rotating the function $y=x^{3}-1$ around the line $x=1$ on the interval $[0,1]$ .

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Problem 262

Determine the function $y=A\cos(Bx-C)+D$ , where $B>0$ and $-\pi<C<\pi$ , that represents the graph with quarter points $(-\pi/2,-2),(-\pi/4,3),(0,8),(\pi/4,3),(\pi/2,-2)$ .

See Solution

Problem 263

Find the equation relating the derivatives of the length, width, and area of a rectangle with $l=18$ and $w=13$ .
$\frac{d A}{d t}=13 \frac{d l}{d t}+18 \frac{d w}{d t}$

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Problem 264

Solve $\frac{20}{1+6 e^{-0.9 t}}=10$ for $t$ using logarithms. Round the answer to two decimal places.

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Problem 265

Find the Taylor series of $f(x) = \ln(x)$ at $a = 6$ . Calculate the first 5 coefficients $c_n$ and the interval of convergence.

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Problem 266

Analyze the function $f(x)=-16 x^{2}-8 x+3$ to determine its critical points, local extrema, and domain/range.

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Problem 267

Find the t-values that make the area under the t-distribution curve between -t and t equal to 0.90, given 20 degrees of freedom.

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Problem 268

Find the range of the quadratic function $y=-x^{2}-2x+3$ where $x$ is bounded.

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Problem 269

What is the value of $(256 \cdot 64)^{\frac{1}{4}}$ ? Select the correct answer from the options.

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Problem 270

Evaluate the indefinite integral of $\frac{3}{2 e^{4 x}}$ with respect to $x$ .

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Problem 271

Find the indefinite integral of $\tan \theta$ with respect to $\theta$ .

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Problem 272

Solve the exponential equation $50 e^{0.035 x} = 200$ for $x$ .

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Problem 273

Given function $q(x) = \frac{1}{x^2 - 9}$ , find $q(y + 3)$ .

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Problem 274

Sketch the graph of the function $k(x)=-\sqrt[3]{x}$ by creating a table of values.

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Problem 275

When showing $\lim_{x\to 0} 4x+5=5$ with $\varepsilon=0.2$ , which $\delta$ -values work? Select all that apply: $\delta=0.0025$ , $\delta=0.016666666666667$ , $\delta=0.05$ , $\delta=0.1$ .

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Problem 276

Determine if $y = \sqrt{1 - x^2}$ represents $y$ as a function of $x$ .

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Problem 277

Evaluate the difference quotient $\frac{f(3+h)-f(3)}{h}$ for $f(x)=4+2x-x^2$ . Simplify the result.

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Problem 278

Determine the behavior of $f(x)$ given that $f'(x)$ is positive, decreasing, and concave down.

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Problem 279

Determine if statement is true or false: If function is positive at $x=a$ , then its derivative is also positive at $x=a$ . Choose correct answer: A) True, sign of rate of change matches sign of value. B) True, derivatives of increasing functions are positive. C) False, sign of rate of change is opposite sign of value. D) False, derivative gives rate of change, not value.

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Problem 280

Find $f(3+h)$ when $f(x) = \frac{1}{x+2}$ .

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Problem 281

Find $f(67.86)$ using the function $f(x)=0.62 \sqrt{53.14+x}$ . Write the answer as a decimal or whole number.

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Problem 282

Find the natural logarithm of 20.

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Problem 283

Differentiate the function $y = \frac{1}{p+k e^{p}}$ and find $y'(p)$ .

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Problem 284

Analyze the symmetry of the function $f(x) = x \cdot e^{x}$ .

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Problem 285

Berechnen Sie die Ableitung der Funktion $f(x) = \sin(6x)$ .

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Problem 286

Find the number of real zeros, end behavior, and max turns of the cubic function $f(x) = -x^3$ .

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Problem 287

Calculate the cube of $5.00 \times 10^{11}$ and express the result in scientific notation.

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Problem 288

Determine the horizontal and vertical asymptotes of the function $\frac{e^{x}}{2 x}$ .

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Problem 289

Determine the end behavior of the graph of the function $f(x) = -3 x^{4} - 6 x^{2} + 9 x + 2$ .

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Problem 290

Find $m$ and determine the $x$ , $y$ intercepts of $f(x)=x^{3}-5x^{2}+mx+9$ , where $(x+1)$ is a factor. Find the stationary points and concavity, then sketch $f(x)$ .

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Problem 291

Plane's landing height is $22,000+(-480 t)$ ft, where $t$ is time in minutes. Estimate the landing time to the nearest hundredth minute.

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Problem 292

Rewrite the exponential function $f(x) = 10^x$ using the natural exponential base $e$ .

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Problem 293

Find the derivative of $y=4u^4+2$ with respect to $x$ , where $u=3\sqrt{x}$ , using the chain rule: $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ .

See Solution

Problem 294

Use a linear approximation to estimate $(3.01)^{3}$ .

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Problem 295

Determine if the piecewise function $f(x) = \begin{cases} x^2 - 4x + 13, & x \leq -4 \\ -12x - 3, & x > -4 \end{cases}$ is differentiable, continuous, both, or neither at the point where the rule changes.

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Problem 296

Find the maximum or minimum value of $f(x) = 3x^2 - 27x + 25$ . If no real solution, enter DNE. Round to 2 decimal places if needed.

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Problem 297

Find the square root parent function from the given options: A. $F(x)=\sqrt{x}$ B. $F(x)=\sqrt{x}+1$ C. $F(x)=x$ D. $F(x)=x^{2}$

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Problem 298

Find the value of $p(5)$ where $p(x) = x^3 - x$ .

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Problem 299

Find the derivative $\frac{d y}{d x}$ of $y=9+\log_{10}(3/x)$ and evaluate it at $x=2$ . Round the answer to three decimal places.

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Problem 300

Determine the concavity of the quadratic function $y = -2(x - 1)^2 + 8$ .

See Solution

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Calculus

Problem 201

Differentiate the following expressions, leaving answers in simplified form with positive exponents: a) g(x)=(x2+4)3g(x)=(x^2+4)^3g(x)=(x2+4)3 (2 marks) b) y=x4(1+4x2)y=x^4(\sqrt{1+4x^2})y=x4(1+4x2​) (4 marks)

Problem 202

Evaluate the integral ∫x3+x154x4+x16−1dx\int \frac{x^{3}+x^{15}}{\sqrt{4 x^{4}+x^{16}-1}} d x∫4x4+x16−1​x3+x15​dx using a substitution, and find the constant of integration.

Problem 203

Find the time(s) when a ball thrown from 5 ft with 23 ft/s upward velocity reaches 13 ft. h=5+23t−16t2h=5+23t-16t^2h=5+23t−16t2, round to nearest hundredth.

Problem 204

Tìm giá trị nhỏ nhất của hàm số y=x+9−x2y=x+\sqrt{9-x^{2}}y=x+9−x2​.

Problem 205

Evaluate the indefinite integral ∫dx9x2−4\int \frac{dx}{\sqrt{9x^2 - 4}}∫9x2−4​dx​.

Problem 206

Find the equation of the normal line to the function f(x)f(x)f(x) at x=1x=1x=1 given f(1)=2f(1)=2f(1)=2, f′(1)=4f'(1)=4f′(1)=4, and f′(2)=1f'(2)=1f′(2)=1.

Problem 207

Find z(x0=1)z(x_0=1)z(x0​=1) and z′(x0=1)z'(x_0=1)z′(x0​=1) for x−8⋅z2=9x^{-8} \cdot z^2 = 9x−8⋅z2=9. Round answers to 2 decimal places.

Problem 208

Find the value of yyy when x=0x=0x=0 in the equation y=ex+1y=e^{x}+1y=ex+1.

Problem 209

Find the slope of the oblique asymptote given by the function f(x)=(ax2+bx+c)/(dx+e)f(x) = (ax^2 + bx + c) / (dx + e)f(x)=(ax2+bx+c)/(dx+e).

Problem 210

Find sss so g(x)=sx2+3x+3g(x) = s x^2 + 3x + 3g(x)=sx2+3x+3 has a minimum at x=−1x = -1x=−1.

Problem 211

Problem 212

Find an exponential function with horizontal asymptote y=−9y=-9y=−9 and passing through points (0,−8)(0,-8)(0,−8) and (1,3)(1,3)(1,3). f(x)=a⋅bxf(x)=a\cdot b^xf(x)=a⋅bx

Problem 213

Estimate the area under f(x)=1x+1f(x)=\frac{1}{x+1}f(x)=x+11​ on [3,5][3,5][3,5] using 8 right-endpoint rectangles. Repeat with left endpoints. RnR_{n}Rn​ and LnL_{n}Ln​ formulas.

Problem 214

Problem 215

Find the 6th term in the Taylor series expansion of f(x)=3e4xf(x)=3e^{4x}f(x)=3e4x around x=4x=4x=4.

Problem 216

Find the antiderivatives of f(x)=−8sec⁡2xf(x)=-8 \sec^2 xf(x)=−8sec2x and verify the solution by taking the derivative.

Problem 217

Estimate the value of yyy using the cubic model y=10x3−12xy=10x^3-12xy=10x3−12x when x=3x=3x=3.

Problem 218

Find the domain of the function f(x)=log⁡6(−5x+4)f(x) = \log_6(-5x+4)f(x)=log6​(−5x+4).

Problem 219

Solve the differential equation 4et⋅(et−1)=04 e^{t} \cdot(e^{t}-1)=04et⋅(et−1)=0 for the value of ttt.

Problem 220

Evaluate f∘g∘h(16)f \circ g \circ h(16)f∘g∘h(16) where f(x)=x2+9f(x) = x^2 + 9f(x)=x2+9, g(x)=x−5g(x) = x - 5g(x)=x−5, and h(x)=xh(x) = \sqrt{x}h(x)=x​.

Problem 221

Find the particular solution form using the method of undetermined coefficients for y′′′+4y′′−5y=xex+11xy'''+ 4y'' - 5y = xe^x + 11xy′′′+4y′′−5y=xex+11x. The form is yp(x)=Axex+Bxy_p(x) = A x e^x + B xyp​(x)=Axex+Bx.

Problem 222

Find the inverse function f−1(x)f^{-1}(x)f−1(x) of f(x)=9+5x−3f(x)=9+\sqrt{5x-3}f(x)=9+5x−3​.

Problem 223

Select the correct statement about the radical function f(x)=3−x3+2f(x)=3 \sqrt[3]{-x}+2f(x)=33−x​+2. A) Strictly increasing B) Strictly decreasing C) Increasing and decreasing D) Constant

Problem 224

Calcul de l'intégrale de f(x)=x2−2xf(x) = \sqrt{x^2 - 2x}f(x)=x2−2x​ sur [2,5][2, 5][2,5] par la méthode de Simpson avec 6 sous-intervalles. Donnez l'approximation à 3 décimales près.

Problem 225

Find a polynomial function ggg where lim⁡x→−∞g(x)=−∞\lim_{x\to-\infty} g(x) = -\inftylimx→−∞​g(x)=−∞. Which expression could define g(x)g(x)g(x)?

Problem 226

Determine the end behavior of f(x)=log⁡(x)f(x) = \log(x)f(x)=log(x) as xxx approaches ∞\infty∞.

Problem 227

Find the function y=f(u)y=f(u)y=f(u) and u=g(x)u=g(x)u=g(x), then compute dydx\frac{dy}{dx}dxdy​ as a function of xxx, where y=cot⁡(5x−6)y=\cot(5x-6)y=cot(5x−6).

Problem 228

Find the equation of the normal line to the curve y=2x2−4x+1y=2x^2-4x+1y=2x2−4x+1 at the point (2,1)(2,1)(2,1) in the form y=mx+cy=mx+cy=mx+c.

Problem 229

Find the simplified difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ and complete the table for f(x)=x2f(x)=x^2f(x)=x2, x=5x=5x=5, and various hhh values.

Problem 230

Find the limit of x+4\sqrt{x+4}x+4​ as xxx approaches 0.

Problem 231

Find the relative maximum point of f(x)=−x3+3x2−4f(x) = -x^3 + 3x^2 - 4f(x)=−x3+3x2−4 using f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x).

Problem 232

Find the domain, limits, and graph of the piecewise function f(x)=25−x2f(x) = \sqrt{25 - x^2}f(x)=25−x2​ for 0≤x<50 \leq x < 50≤x<5, f(x)=5f(x) = 5f(x)=5 for 5≤x<105 \leq x < 105≤x<10, and f(x)=10f(x) = 10f(x)=10 for x=10x = 10x=10.

Problem 233

Problem 234

Find the derivative of y=sin⁡n(x)cos⁡n(x)y = \sin^n(x)\cos^n(x)y=sinn(x)cosn(x) for n>2n>2n>2.

Problem 235

Evaluate the indefinite integral of 12x3−12x5+x12x^3 - 12x^5 + \sqrt{x}12x3−12x5+x​.

Problem 236

Graph f(x)=∣4x+12∣f(x)=|4x+12|f(x)=∣4x+12∣ and find lim⁡x→−3f(x)\lim_{x\to-3}f(x)limx→−3​f(x) and lim⁡x→−5f(x)\lim_{x\to-5}f(x)limx→−5​f(x). If necessary, state that the limit does not exist.

Problem 237

Find the equation of the tangent line to y=x2−3y=x^2-3y=x2−3 at x=4x=4x=4. A. y=8x−19y=8x-19y=8x−19 B. y=8x−35y=8x-35y=8x−35 C. y=4x−19y=4x-19y=4x−19 D. y=8x−38y=8x-38y=8x−38

Problem 238

Evaluate the limit of x2+2x+4x^2 + \frac{2}{x+4}x2+x+42​ as xxx approaches -4 from the left. Determine the limits of f(x)f(x)f(x) and g(x)g(x)g(x) as xxx approaches ccc, and evaluate the limits of their sum, product, and ratio.

Problem 239

Find the value of f(g(256))f(g(256))f(g(256)) and g(f(−4))g(f(-4))g(f(−4)) for the functions f(x)=2xf(x)=2^{x}f(x)=2x and g(x)=log⁡4xg(x)=\log_{4} xg(x)=log4​x.

Problem 240

Evaluate 61ln⁡e61\sqrt{61} \ln e^{\sqrt{61}}61​lne61​ without a calculator and simplify the result.

Problem 241

Evaluate the integral ∫cos⁡3xcos⁡3xdx\int \cos 3x \cos 3x dx∫cos3xcos3xdx and select the correct solution from the given options.

Differentiate the following expressions, leaving answers in simplified form with positive exponents: a) $g(x)=(x^2+4)^3$ (2 marks) b) $y=x^4(\sqrt{1+4x^2})$ (4 marks)

Evaluate the integral $\int \frac{x^{3}+x^{15}}{\sqrt{4 x^{4}+x^{16}-1}} d x$ using a substitution, and find the constant of integration.

Find the time(s) when a ball thrown from 5 ft with 23 ft/s upward velocity reaches 13 ft. $h=5+23t-16t^2$ , round to nearest hundredth.

Tìm giá trị nhỏ nhất của hàm số $y=x+\sqrt{9-x^{2}}$ .

Evaluate the indefinite integral $\int \frac{dx}{\sqrt{9x^2 - 4}}$ .

Find the equation of the normal line to the function $f(x)$ at $x=1$ given $f(1)=2$ , $f'(1)=4$ , and $f'(2)=1$ .

Find $z(x_0=1)$ and $z'(x_0=1)$ for $x^{-8} \cdot z^2 = 9$ . Round answers to 2 decimal places.

Find the value of $y$ when $x=0$ in the equation $y=e^{x}+1$ .

Find the slope of the oblique asymptote given by the function $f(x) = (ax^2 + bx + c) / (dx + e)$ .

Find $s$ so $g(x) = s x^2 + 3x + 3$ has a minimum at $x = -1$ .

Find an exponential function with horizontal asymptote $y=-9$ and passing through points $(0,-8)$ and $(1,3)$ . $f(x)=a\cdot b^x$

Estimate the area under $f(x)=\frac{1}{x+1}$ on $[3,5]$ using 8 right-endpoint rectangles. Repeat with left endpoints. $R_{n}$ and $L_{n}$ formulas.

Find the 6th term in the Taylor series expansion of $f(x)=3e^{4x}$ around $x=4$ .

Find the antiderivatives of $f(x)=-8 \sec^2 x$ and verify the solution by taking the derivative.

Estimate the value of $y$ using the cubic model $y=10x^3-12x$ when $x=3$ .

Find the domain of the function $f(x) = \log_6(-5x+4)$ .

Solve the differential equation $4 e^{t} \cdot(e^{t}-1)=0$ for the value of $t$ .

Evaluate $f \circ g \circ h(16)$ where $f(x) = x^2 + 9$ , $g(x) = x - 5$ , and $h(x) = \sqrt{x}$ .

Find the particular solution form using the method of undetermined coefficients for $y'''+ 4y'' - 5y = xe^x + 11x$ . The form is $y_p(x) = A x e^x + B x$ .

Find the inverse function $f^{-1}(x)$ of $f(x)=9+\sqrt{5x-3}$ .

Select the correct statement about the radical function $f(x)=3 \sqrt[3]{-x}+2$ . A) Strictly increasing B) Strictly decreasing C) Increasing and decreasing D) Constant

Calcul de l'intégrale de $f(x) = \sqrt{x^2 - 2x}$ sur $[2, 5]$ par la méthode de Simpson avec 6 sous-intervalles. Donnez l'approximation à 3 décimales près.

Find a polynomial function $g$ where $\lim_{x\to-\infty} g(x) = -\infty$ . Which expression could define $g(x)$ ?

Determine the end behavior of $f(x) = \log(x)$ as $x$ approaches $\infty$ .

Find the function $y=f(u)$ and $u=g(x)$ , then compute $\frac{dy}{dx}$ as a function of $x$ , where $y=\cot(5x-6)$ .

Find the equation of the normal line to the curve $y=2x^2-4x+1$ at the point $(2,1)$ in the form $y=mx+c$ .

Find the simplified difference quotient $\frac{f(x+h)-f(x)}{h}$ and complete the table for $f(x)=x^2$ , $x=5$ , and various $h$ values.

Find the limit of $\sqrt{x+4}$ as $x$ approaches 0.

Find the relative maximum point of $f(x) = -x^3 + 3x^2 - 4$ using $f'(x)$ and $f''(x)$ .

Find the domain, limits, and graph of the piecewise function $f(x) = \sqrt{25 - x^2}$ for $0 \leq x < 5$ , $f(x) = 5$ for $5 \leq x < 10$ , and $f(x) = 10$ for $x = 10$ .

Find the derivative of $y = \sin^n(x)\cos^n(x)$ for $n>2$ .

Evaluate the indefinite integral of $12x^3 - 12x^5 + \sqrt{x}$ .

Graph $f(x)=|4x+12|$ and find $\lim_{x\to-3}f(x)$ and $\lim_{x\to-5}f(x)$ . If necessary, state that the limit does not exist.

Find the equation of the tangent line to $y=x^2-3$ at $x=4$ . A. $y=8x-19$ B. $y=8x-35$ C. $y=4x-19$ D. $y=8x-38$

Evaluate the limit of $x^2 + \frac{2}{x+4}$ as $x$ approaches -4 from the left. Determine the limits of $f(x)$ and $g(x)$ as $x$ approaches $c$ , and evaluate the limits of their sum, product, and ratio.

Find the value of $f(g(256))$ and $g(f(-4))$ for the functions $f(x)=2^{x}$ and $g(x)=\log_{4} x$ .

Evaluate $\sqrt{61} \ln e^{\sqrt{61}}$ without a calculator and simplify the result.

Evaluate the integral $\int \cos 3x \cos 3x dx$ and select the correct solution from the given options.

Find the derivative of $3^x$ with respect to $x$ . Type $\ln(x)$ for the natural logarithm function.

Calculate the integral $\int 2 \sqrt[3]{x} \, dx$ and express the result in simplest form.

Calculate the integral of $-2 x^{-2} + 2 - 4 x$ and simplify the result.

Find the limit of $\frac{\tan(2x)}{x\cos(x)}$ as $x$ approaches 0.

Solve for $x$ in the equation $5 \ln x = -15$ . Select the correct choice: A. $x = e^{-3}$ , or B. The solution is not a real number.

Find the critical points of the polynomial function $f(x) = x^4 + 4x^3 - 9$ .

Find the area between the cubic curves $y=x^{3}-13x^{2}+30x$ and $y=-x^{3}+13x^{2}-30x$ .

Sketch the graph of $f(x)=-e^{x-3}$ using transformations of $y=e^x$ . Find the domain, range, $y$ -intercept, and horizontal asymptote.

Find the derivative of $\sin(g(x))$ evaluated at $x=8$ , given that $g(8)=3$ and $g'(8)=9$ . Round the answer to three decimal places.

Find the range of $g(x) = -8^x$ . (Answer in interval notation)

Differentiate the function $f(t) = \frac{\sec t}{-1 + \sec t}$ and find the derivative $f'(t)$ .

Find the intervals where the function $f(x) = (9 - 6x)e^x$ is concave up and concave down.

Find the derivative of the inverse function of $f(x) = 4x - 9$ at the point $(-13, -1)$ on the inverse function's graph.

Estimate the errors in approximating the integral $\int_{2}^{4} \frac{1}{(x-1)^{2}} dx$ using the Trapezoidal Rule and Simpson's Rule with $n=4$ .

Find the value of $y$ if $x = e^y$ .

Find the domain of the inverse function $f^{-1}$ given that the domain of $f$ is $[5, \infty)$ and its range is $[-9, \infty)$ .
The domain of $f^{-1}$ is $[-9, \infty)$ .

Find the derivative of $t(x)=0.08 e^{x} x^{2}$ evaluated at $x=5$ , rounded to the nearest tenth.

Find the domain of the function $g(x) = \sqrt{8x+1}$ .

Find the volume of the solid formed by rotating the function $y=x^{3}-1$ around the line $x=1$ on the interval $[0,1]$ .

Find the equation relating the derivatives of the length, width, and area of a rectangle with $l=18$ and $w=13$ .
$\frac{d A}{d t}=13 \frac{d l}{d t}+18 \frac{d w}{d t}$

Solve $\frac{20}{1+6 e^{-0.9 t}}=10$ for $t$ using logarithms. Round the answer to two decimal places.

Find the Taylor series of $f(x) = \ln(x)$ at $a = 6$ . Calculate the first 5 coefficients $c_n$ and the interval of convergence.

Analyze the function $f(x)=-16 x^{2}-8 x+3$ to determine its critical points, local extrema, and domain/range.

Find the range of the quadratic function $y=-x^{2}-2x+3$ where $x$ is bounded.

What is the value of $(256 \cdot 64)^{\frac{1}{4}}$ ? Select the correct answer from the options.

Evaluate the indefinite integral of $\frac{3}{2 e^{4 x}}$ with respect to $x$ .

Find the indefinite integral of $\tan \theta$ with respect to $\theta$ .

Solve the exponential equation $50 e^{0.035 x} = 200$ for $x$ .

Given function $q(x) = \frac{1}{x^2 - 9}$ , find $q(y + 3)$ .

Sketch the graph of the function $k(x)=-\sqrt[3]{x}$ by creating a table of values.