Calculus

Problem 301

Evaluate the quotient $\left(5.44 \times 10^{-18}\right) \div\left(6.8 \times 10^{-9}\right)$ and express the result in scientific notation.

See Solution

Problem 302

Find the regions of increasing and decreasing for the function $y=\frac{2x+7}{2x}$ where $x \neq 0$ .

See Solution

Problem 303

Solve the equation $7 e^{3 x}-4=14$ using natural logarithms. Round the solution to the nearest thousandth.

See Solution

Problem 304

Find the quadratic equation of a rocket's trajectory given its maximum height of $1280$ feet and landing point at $(16,0)$ in $8$ seconds.

See Solution

Problem 305

Find the derivative of the functions $f(x)=2x^2$ , $f(x)=2x$ , $f(x)=5$ , $f(x)=-x^2$ , $f(x)=2x+3$ , and $f(x)=ax^2$ .

See Solution

Problem 306

Find the absolute extreme values of $f(x) = \frac{12x^3}{3} + 16x^2 - 12x$ on the interval $[-4, 1]$ . Select the correct choice: A. The absolute maximum is $\square$ at $x = \square$ . B. There is no absolute maximum on the given interval.

See Solution

Problem 307

Find the derivative of the inverse function $f^{-1}(x)$ where $f(x)=\frac{1}{(1-x)^{2}}$ and $x>1$ , evaluated at $x=\frac{1}{4}$ .

See Solution

Problem 308

Calculate the specified partial derivatives for the following functions: (a) $f(x, y)=x y^{3} e^{y}, \quad f_{y^{2}}^{\prime \prime}=\frac{\partial^{2} f}{\partial y^{2}}$ (b) $f(p, q)=3 p^{3} q^{2}, \quad f_{p^{2}}^{\prime \prime}=\frac{\partial^{2} f}{\partial p^{2}}$ (c) $f(k, l)=5 \sqrt{k} l^{3}, \quad f_{kl}^{\prime \prime}=\frac{\partial^{2} f}{\partial k \partial l}$

See Solution

Problem 309

Find the approximate value of $y=\sqrt{4+\sin x}$ at $x=0.12$ .

See Solution

Problem 310

Find the limit of the given rational expression as $x$ approaches 3.

See Solution

Problem 311

Determine constant $c$ so that $f(x) = c e^{-(2x+3)}$ is a valid probability distribution over $0.621 > x > 0.4379$ .

See Solution

Problem 312

Find the indefinite integral of $\cos^2 3x$ .

See Solution

Problem 313

Evaluate the integral $\int x \sec^{-1} x dx$ for $x \leq -1$ .

See Solution

Problem 314

Find the area of the region between $y=3\sin(x)$ and $y=4\cos(x)$ from $x=0$ to $x=0.6\pi$ . The region consists of two parts.

See Solution

Problem 315

Calculate the integral $\int x \ln x d x$ . (3 marks for calculation, 3 marks for explanations, 6 marks total.)

See Solution

Problem 316

Find the second derivative $f''(1)$ of a differentiable function $f$ satisfying $f'(x) + 3f(x^2) = f^2(x)$ for $x > 0$ and $f(1) = 5$ .

See Solution

Problem 317

Find the function $k(-x)$ given $k(x)=-8 x^{9}-7 x$ .

See Solution

Problem 318

Find the range of $f(x)=x^2+3$ when $x>0$ .

See Solution

Problem 319

Find the derivatives of $h(x) = f(x)g(x)$ at $x = 3$ and $h(x) = f(x)/g(x)$ at $x = 2$ , given a table of $f(x), f'(x), g(x), g'(x)$ . Also, find the points where the tangent line of $f(x) = (2x^2 + 10)/(x + 2)$ is horizontal.

See Solution

Problem 320

Find the point(s) on the graph of $f(x)=3x^2-5x$ with tangent lines parallel to $y=7x+5$ . The point(s) is/are $($ x, y $)$ .

See Solution

Problem 321

(1 point) Evaluate the definite integral $\int_{1}^{3} 16 x^{2} d x$ using: a) Trapezoid Rule with $n=4$ , b) Simpson's Rule with $n=4$ , and c) find the exact value.

See Solution

Problem 322

Find the left Riemann sum $L_n$ , right Riemann sum $R_n$ , and their average for $\int_1^5 (x^2 + 4) dx$ with $n=4$ .

See Solution

Problem 323

Find the integral of $2x f(x^2) + 8$ from $x=2$ to $x=4$ given that $f(x)$ is continuous and $\int_{4}^{16} f(x) dx = 19$ .

See Solution

Problem 324

Find the derivative $\frac{d y}{d x}$ of the equation $6 y^{2} - (x - 1) y = 5$ in simplified factored form.

See Solution

Problem 325

Find the range of the function $f(x) = \sqrt{x-2}$ on its domain.

See Solution

Problem 326

Find the indefinite integral of $3x\sin x^2$ .

See Solution

Problem 327

Approximate $e^{4}$ and round the answer to the nearest tenth.

See Solution

Problem 328

Find the maximum height of a vertically moving body with $s=-\frac{1}{2}gt^2+v_0t+s_0, g>0$ . Which expression gives the correct max height: $\frac{v_0^2}{2g}+s_0$ , $\frac{v_0}{g}+s_0$ , $\frac{v_0^2}{g}+s_0$ , $\frac{v_0}{2g}+s_0$ ?

See Solution

Problem 329

Find the derivative of $y=3 \sec x \tan x$ with respect to $x$ .

See Solution

Problem 330

Find the integral of $e^{x}/x$ from 1 to 2, given $F(x)=\int_{0}^{x} e^{e^{t}} dt$ .

See Solution

Problem 331

Find the value of the third derivative of $f(x)$ at $x=-1$ , given the degree 3 Taylor polynomial $P(x)=8-\frac{9}{10}(x+1)+\frac{8}{7}(x+1)^{2}-\frac{10}{3}(x+1)^{3}$ .

See Solution

Problem 332

Evaluate the natural logarithm of $e^{7}$ .

See Solution

Problem 333

Approximate the integral of $f(x) = 2 - x^2$ from 0 to 1 using Riemann sums. Lisa calculates the upper sum using the maximum function value, while Theo calculates the lower sum using the minimum function value. Compare the results for 5 and 500 rectangles.

See Solution

Problem 334

Determine the intersection point of the tangents at points $A$ and $B$ on the graph of the function $f(x) = -\frac{1}{2}x^{2} - 2x$ , where $A(-1/1.5)$ and $B(1/2.5)$ .

See Solution

Problem 335

Determine if $y=\frac{1}{4} e^{-3 x}$ represents exponential growth or decay. Identify the graph.

See Solution

Problem 336

Find the derivative $\frac{d y}{d x}$ when $x=\sqrt{y}$ .

See Solution

Problem 337

Find the critical points of the function $f(\theta) = 8 \sec \theta + 4 \tan \theta$ for $0 < \theta < 2\pi$ .

See Solution

Problem 338

Find a function $f(x)$ with derivative $f'(x) = 2$ that passes through the point $(0, 3)$ .

See Solution

Problem 339

Find the derivative of $f(x)=\sqrt{7x+4}$ using the limit definition. Then find the equation of the tangent line at $x=3$ .

See Solution

Problem 340

Find the limit of the piecewise function $g(x)$ as $x$ approaches 2 from the right.

See Solution

Problem 341

Find the values of $\ln 35.2$ and $\log \frac{4}{5}$ .

See Solution

Problem 342

Find the term independent of $x$ in the expansion of $\left(\frac{1}{x}+x^{2}\right)^{9}$ .

See Solution

Problem 343

Find the value of $s(r(4))$ where $r(x) = 2x - 2$ and $s(x) = -x^2$ .

See Solution

Problem 344

Use leading coefficient test to determine if $y \to \infty$ or $y \to -\infty$ as $x \to -\infty$ for $y = 7x^5 - 6x^4$ .

See Solution

Problem 345

Expand the expression $e^{x-\ln x}$ .

See Solution

Problem 346

Determine if the graphs of the function family $f_a(x) = x^4 - ax^2$ have no local maximum for $a \leq 0$ .

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Problem 347

Find the slope of the function $y = |x| + 1$ .

See Solution

Problem 348

Find two functions $f$ and $g$ such that $(f \circ g)(x) = (6x+7)^7$ , where neither function is the identity function.

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Problem 349

Compute $\Delta y$ and $dy$ for $y=\sqrt{x-1}$ , $x=2$ , $\Delta x=0.8$ . Round answers to three decimal places. Sketch a diagram showing $dx$ , $dy$ , and $\Delta y$ .

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Problem 350

Find $b$ in $\ln(25b) = \square$ .

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Problem 351

Find the values of $x$ where the function $f(x)$ is negative, based on the graph.

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Problem 352

Find the rate of change of the base of a triangle with altitude 12 cm and area 97 sq cm, given the altitude is increasing at 2.5 cm/min and the area at 3.5 sq cm/min.

See Solution

Problem 353

Find the exponential function $g(t) = ab^t$ given $g(10) = 95$ and $g(30) = 27$ . Round the solution to four decimal places.

See Solution

Problem 354

Find the derivative of $f(x) = -\sqrt{x^3} - \frac{\sqrt{x}}{2}$ and evaluate it at $x = 3$ . Express the answer as a simplified radical fraction.

See Solution

Problem 355

Find the limit of $(f(x+h) - f(x))/h$ as $h$ approaches 0, given $f(x) = x^2 - 8$ .

See Solution

Problem 356

Determine the $x$ - and $y$ -coordinates of the inflection point of the function $f(x)=\frac{e^{x}}{1+e^{x}}$ .

See Solution

Problem 357

Given functions $f(x) = 0.5x^2$ and $g(x) = 3x^3 + 1$ . Find the difference quotient for each interval: a) $[0; 2]$ , b) $[-1; 3]$ , c) $[-1; 1]$ , d) $[-2; -1]$ .

See Solution

Problem 358

Determine if $E=mc^2$ is a power function. If so, state the power and constant of variation.

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Problem 359

Describe the end behavior of the function $y=-|2x+2|+3$ . As $x \rightarrow-\infty, y \rightarrow+\infty$ ; As $x \rightarrow+\infty, y \rightarrow-\infty$ .

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Problem 360

Given the function $V(x) = (x - 4)^2 + 6, x \geq 4$ , find the minimum value of the function.

See Solution

Problem 361

Differentiate the function $y=\frac{1}{x+\frac{1}{x+1}}$ with respect to $x$ .

See Solution

Problem 362

Finde kritische Stelle und Grenzwerte einer Funktion $f$ mit vertikaler Asymptote bei $x=1$ .

See Solution

Problem 363

Evaluate $N(0)$ and $N(18)$ for $N(t) = \frac{12.80}{0.02 + 0.53^{t}}$ . Round answers to 2 decimal places.

See Solution

Problem 364

Find the derivative of $f(x) = x - x^2$ using the limit definition of the derivative.

See Solution

Problem 365

Find the absolute extrema and their $x$ -values for $f(x)=2x^3+8x^2+8x-2$ on $[-5,0]$ .

See Solution

Problem 366

Find the critical point and determine the behavior of the function $y=x^{7/2}-4x^2$ for $x>0$ . Critical point: $c=$

See Solution

Problem 367

Simplify $\frac{e^{x} + 7e^{6x}}{e^{x}}$ before finding its derivative.

See Solution

Problem 368

Find the domain, vertical asymptote(s), and horizontal asymptote of $f(x)=\frac{1}{x+2}$ . Domain: $x\neq -2$ , Vertical Asymptote(s): $x=-2$ , Horizontal Asymptote: $y=0$ .

See Solution

Problem 369

Find the $y$ -values of the function $f(x) = -3e^x$ to two decimal places when $x = -2, -1, 0, 1, 2$ .

See Solution

Problem 370

Find the time when the drug concentration $C(t)=\frac{0.7 t}{t^{2}+49}$ is maximized, accurate to at least 2 decimal places.

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Problem 371

Find the logarithmic function $f(t) = a + \log_b(t)$ passing through points $(4, -0.5)$ and $(16, 0)$ .

See Solution

Problem 372

Determine the function resulting from horizontally compressing $y=f(x)$ by $1/3$ and vertically translating the graph 6 units down.

See Solution

Problem 373

Determine the derivative of the function $f(x) = \frac{1}{2} x^{3}$ .

See Solution

Problem 374

Solve for $x$ to two decimal places using the exponential equation $700=500(1.04)^{x}$ .

See Solution

Problem 375

Drug concentration function $K(t) = \frac{8t}{t^2 + 4}$ for $0 < t < \infty$ . a. Find intervals where $K(t)$ is increasing. b. Find intervals where $K(t)$ is decreasing.

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Problem 376

Solve the equation $8 x \ln x + x = 0$ and round the answer to 3 decimal places.

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Problem 377

Find the roots of the derivative $f'_a(t) = 1.5t^2 - 3(a+1)t + 6a$ for a general 'a'. Solve $1.5t^2 - 3(a+1)t + 6a = 0$ .

See Solution

Problem 378

Use implicit differentiation to find the derivative $\frac{d p}{d x}$ for the demand equation $p^{2} + p + 7x = 200$ .

See Solution

Problem 379

Evaluate $e^2$ , $e^{-2}$ , and $e^{1/2}$ using a calculator, rounding to three decimal places.

See Solution

Problem 380

Estimate $\Delta y$ for $y=\sin(5x)$ using linear approximation when $\Delta x=0.3$ at $x=0$ . Find the percentage error.

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Problem 381

Find the domain and range of $y=\log_{4}(4+2x)$ . Explain how to solve without graphing.

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Problem 382

Find the values of a, b, and c such that $7^{9/4} = \sqrt[c]{a^{b}}$ .

See Solution

Problem 383

The exponential function $f(x)=561(1.026)^{x}$ models a country's population $f(x)$ in millions, $x$ years after 1974. Find the population in 1974, 2001, 2028, and 2055, and describe the population growth pattern.

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Problem 384

Find the function with a different asymptote from the others: $q(x)=6 \cdot \log _{5}(x-2)+3$ , $b(x)=4 \cdot \log _{5}(x-2)+3$ , $c(x)=4 \cdot \log _{s}(x+1)+3$ , $d(x)=4 \cdot \log _{5}(x-2)-7$ .

See Solution

Problem 385

Find the natural logarithm of the 10th root of e without using a calculator. Solve for the 10th root of e.

See Solution

Problem 386

Find the range of the function $f(x) = \log_2(x+2) + 1$ where $x > -2$ and $-2 < y < 4$ .

See Solution

Problem 387

Find variables in $V=\pi r^{2} h$ to create linear or quadratic relationships. Explain your choices.

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Problem 388

Estimate $\sqrt{85}$ using linear approximation. Choose a value of $a$ to minimize the error. Round the approximation to three decimal places.

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Problem 389

Approximate $e$ by substituting large $n$ into $\left(1+\frac{1}{n}\right)^{n}$ .

See Solution

Problem 390

Expand $(1-2x)^{1/2}$ as a power series in $x$ up to and including $x^3$ , simplifying coefficients, where $-1/2<x<1/2$ .

See Solution

Problem 391

Sketch the graph of $y=x^{4}-8x^{3}+16x^{2}-2$ and summarize key points in a table.

See Solution

Problem 392

Find the value of $a$ if $7^{a}=\sqrt[8]{7^{3}}$ .

See Solution

Problem 393

Solve the equation $4 \sqrt{x} + 6 = 7$ for $x$ .

See Solution

Problem 394

Use the quotient rule to simplify $\frac{p^{2} q^{3}}{p q^{2}}$ , where all bases are not equal to 0.

See Solution

Problem 395

What operation undoes $X^{2}$ ? a) Divide by 2 b) Logit? c) Square root? d) Square it.

See Solution

Problem 396

Solve for $x$ in the equation $\ln (9 x+4)=4.2$ , rounding the final solution to 3 decimal places.

See Solution

Problem 397

Bestimme die Ableitung und Nullstellen der Funktion $y=x^{2}+1$ und skizziere den Graphen.

See Solution

Problem 398

Graph the equation $y=\log_{3}(x+3)+3$ , then find its domain, range, and vertical asymptote.

See Solution

Problem 399

Evaluate $f(x)=\sqrt{x+8}$ at $x=-4$ and simplify.

See Solution

Problem 400

Find the limit of the sequence $\lim _{n \rightarrow \infty} \sqrt[n]{7^{n}-3^{n}}$ .

See Solution

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Calculus

Problem 301

Evaluate the quotient (5.44×10−18)÷(6.8×10−9)\left(5.44 \times 10^{-18}\right) \div\left(6.8 \times 10^{-9}\right)(5.44×10−18)÷(6.8×10−9) and express the result in scientific notation.

Problem 302

Find the regions of increasing and decreasing for the function y=2x+72xy=\frac{2x+7}{2x}y=2x2x+7​ where x≠0x \neq 0x=0.

Problem 303

Solve the equation 7e3x−4=147 e^{3 x}-4=147e3x−4=14 using natural logarithms. Round the solution to the nearest thousandth.

Problem 304

Find the quadratic equation of a rocket's trajectory given its maximum height of 128012801280 feet and landing point at (16,0)(16,0)(16,0) in 888 seconds.

Problem 305

Find the derivative of the functions f(x)=2x2f(x)=2x^2f(x)=2x2, f(x)=2xf(x)=2xf(x)=2x, f(x)=5f(x)=5f(x)=5, f(x)=−x2f(x)=-x^2f(x)=−x2, f(x)=2x+3f(x)=2x+3f(x)=2x+3, and f(x)=ax2f(x)=ax^2f(x)=ax2.

Problem 306

Problem 307

Find the derivative of the inverse function f−1(x)f^{-1}(x)f−1(x) where f(x)=1(1−x)2f(x)=\frac{1}{(1-x)^{2}}f(x)=(1−x)21​ and x>1x>1x>1, evaluated at x=14x=\frac{1}{4}x=41​.

Problem 308

Problem 309

Find the approximate value of y=4+sin⁡xy=\sqrt{4+\sin x}y=4+sinx​ at x=0.12x=0.12x=0.12.

Problem 310

Find the limit of the given rational expression as xxx approaches 3.

Problem 311

Determine constant ccc so that f(x)=ce−(2x+3)f(x) = c e^{-(2x+3)}f(x)=ce−(2x+3) is a valid probability distribution over 0.621>x>0.43790.621 > x > 0.43790.621>x>0.4379.

Problem 312

Find the indefinite integral of cos⁡23x\cos^2 3xcos23x.

Problem 313

Evaluate the integral ∫xsec⁡−1xdx\int x \sec^{-1} x dx∫xsec−1xdx for x≤−1x \leq -1x≤−1.

Problem 314

Find the area of the region between y=3sin⁡(x)y=3\sin(x)y=3sin(x) and y=4cos⁡(x)y=4\cos(x)y=4cos(x) from x=0x=0x=0 to x=0.6πx=0.6\pix=0.6π. The region consists of two parts.

Problem 315

Calculate the integral ∫xln⁡xdx\int x \ln x d x∫xlnxdx. (3 marks for calculation, 3 marks for explanations, 6 marks total.)

Problem 316

Find the second derivative f′′(1)f''(1)f′′(1) of a differentiable function fff satisfying f′(x)+3f(x2)=f2(x)f'(x) + 3f(x^2) = f^2(x)f′(x)+3f(x2)=f2(x) for x>0x > 0x>0 and f(1)=5f(1) = 5f(1)=5.

Problem 317

Find the function k(−x)k(-x)k(−x) given k(x)=−8x9−7xk(x)=-8 x^{9}-7 xk(x)=−8x9−7x.

Problem 318

Find the range of f(x)=x2+3f(x)=x^2+3f(x)=x2+3 when x>0x>0x>0.

Problem 319

Problem 320

Find the point(s) on the graph of f(x)=3x2−5xf(x)=3x^2-5xf(x)=3x2−5x with tangent lines parallel to y=7x+5y=7x+5y=7x+5. The point(s) is/are (((x, y))).

Problem 321

(1 point) Evaluate the definite integral ∫1316x2dx\int_{1}^{3} 16 x^{2} d x∫13​16x2dx using: a) Trapezoid Rule with n=4n=4n=4, b) Simpson's Rule with n=4n=4n=4, and c) find the exact value.

Problem 322

Find the left Riemann sum LnL_nLn​, right Riemann sum RnR_nRn​, and their average for ∫15(x2+4)dx\int_1^5 (x^2 + 4) dx∫15​(x2+4)dx with n=4n=4n=4.

Problem 323

Find the integral of 2xf(x2)+82x f(x^2) + 82xf(x2)+8 from x=2x=2x=2 to x=4x=4x=4 given that f(x)f(x)f(x) is continuous and ∫416f(x)dx=19\int_{4}^{16} f(x) dx = 19∫416​f(x)dx=19.

Problem 324

Find the derivative dydx\frac{d y}{d x}dxdy​ of the equation 6y2−(x−1)y=56 y^{2} - (x - 1) y = 56y2−(x−1)y=5 in simplified factored form.

Problem 325

Find the range of the function f(x)=x−2f(x) = \sqrt{x-2}f(x)=x−2​ on its domain.

Problem 326

Find the indefinite integral of 3xsin⁡x23x\sin x^23xsinx2.

Problem 327

Approximate e4e^{4}e4 and round the answer to the nearest tenth.

Problem 328

Problem 329

Find the derivative of y=3sec⁡xtan⁡xy=3 \sec x \tan xy=3secxtanx with respect to xxx.

Problem 330

Find the integral of ex/xe^{x}/xex/x from 1 to 2, given F(x)=∫0xeetdtF(x)=\int_{0}^{x} e^{e^{t}} dtF(x)=∫0x​eetdt.

Problem 331

Find the value of the third derivative of f(x)f(x)f(x) at x=−1x=-1x=−1, given the degree 3 Taylor polynomial P(x)=8−910(x+1)+87(x+1)2−103(x+1)3P(x)=8-\frac{9}{10}(x+1)+\frac{8}{7}(x+1)^{2}-\frac{10}{3}(x+1)^{3}P(x)=8−109​(x+1)+78​(x+1)2−310​(x+1)3.

Problem 332

Evaluate the natural logarithm of e7e^{7}e7.

Problem 333

Approximate the integral of f(x)=2−x2f(x) = 2 - x^2f(x)=2−x2 from 0 to 1 using Riemann sums. Lisa calculates the upper sum using the maximum function value, while Theo calculates the lower sum using the minimum function value. Compare the results for 5 and 500 rectangles.

Problem 334

Determine the intersection point of the tangents at points AAA and BBB on the graph of the function f(x)=−12x2−2xf(x) = -\frac{1}{2}x^{2} - 2xf(x)=−21​x2−2x, where A(−1/1.5)A(-1/1.5)A(−1/1.5) and B(1/2.5)B(1/2.5)B(1/2.5).

Problem 335

Determine if y=14e−3xy=\frac{1}{4} e^{-3 x}y=41​e−3x represents exponential growth or decay. Identify the graph.

Problem 336

Find the derivative dydx\frac{d y}{d x}dxdy​ when x=yx=\sqrt{y}x=y​.

Problem 337

Find the critical points of the function f(θ)=8sec⁡θ+4tan⁡θf(\theta) = 8 \sec \theta + 4 \tan \thetaf(θ)=8secθ+4tanθ for 0<θ<2π0 < \theta < 2\pi0<θ<2π.

Problem 338

Find a function f(x)f(x)f(x) with derivative f′(x)=2f'(x) = 2f′(x)=2 that passes through the point (0,3)(0, 3)(0,3).

Problem 339

Find the derivative of f(x)=7x+4f(x)=\sqrt{7x+4}f(x)=7x+4​ using the limit definition. Then find the equation of the tangent line at x=3x=3x=3.

Problem 340

Find the limit of the piecewise function g(x)g(x)g(x) as xxx approaches 2 from the right.

Problem 341

Find the values of ln⁡35.2\ln 35.2ln35.2 and log⁡45\log \frac{4}{5}log54​.

Problem 342

Evaluate the quotient $\left(5.44 \times 10^{-18}\right) \div\left(6.8 \times 10^{-9}\right)$ and express the result in scientific notation.

Find the regions of increasing and decreasing for the function $y=\frac{2x+7}{2x}$ where $x \neq 0$ .

Solve the equation $7 e^{3 x}-4=14$ using natural logarithms. Round the solution to the nearest thousandth.

Find the quadratic equation of a rocket's trajectory given its maximum height of $1280$ feet and landing point at $(16,0)$ in $8$ seconds.

Find the derivative of the functions $f(x)=2x^2$ , $f(x)=2x$ , $f(x)=5$ , $f(x)=-x^2$ , $f(x)=2x+3$ , and $f(x)=ax^2$ .

Find the derivative of the inverse function $f^{-1}(x)$ where $f(x)=\frac{1}{(1-x)^{2}}$ and $x>1$ , evaluated at $x=\frac{1}{4}$ .

Find the approximate value of $y=\sqrt{4+\sin x}$ at $x=0.12$ .

Find the limit of the given rational expression as $x$ approaches 3.

Determine constant $c$ so that $f(x) = c e^{-(2x+3)}$ is a valid probability distribution over $0.621 > x > 0.4379$ .

Find the indefinite integral of $\cos^2 3x$ .

Evaluate the integral $\int x \sec^{-1} x dx$ for $x \leq -1$ .

Find the area of the region between $y=3\sin(x)$ and $y=4\cos(x)$ from $x=0$ to $x=0.6\pi$ . The region consists of two parts.

Calculate the integral $\int x \ln x d x$ . (3 marks for calculation, 3 marks for explanations, 6 marks total.)

Find the second derivative $f''(1)$ of a differentiable function $f$ satisfying $f'(x) + 3f(x^2) = f^2(x)$ for $x > 0$ and $f(1) = 5$ .

Find the function $k(-x)$ given $k(x)=-8 x^{9}-7 x$ .

Find the range of $f(x)=x^2+3$ when $x>0$ .

Find the point(s) on the graph of $f(x)=3x^2-5x$ with tangent lines parallel to $y=7x+5$ . The point(s) is/are $($ x, y $)$ .

(1 point) Evaluate the definite integral $\int_{1}^{3} 16 x^{2} d x$ using: a) Trapezoid Rule with $n=4$ , b) Simpson's Rule with $n=4$ , and c) find the exact value.

Find the left Riemann sum $L_n$ , right Riemann sum $R_n$ , and their average for $\int_1^5 (x^2 + 4) dx$ with $n=4$ .

Find the integral of $2x f(x^2) + 8$ from $x=2$ to $x=4$ given that $f(x)$ is continuous and $\int_{4}^{16} f(x) dx = 19$ .

Find the derivative $\frac{d y}{d x}$ of the equation $6 y^{2} - (x - 1) y = 5$ in simplified factored form.

Find the range of the function $f(x) = \sqrt{x-2}$ on its domain.

Find the indefinite integral of $3x\sin x^2$ .

Approximate $e^{4}$ and round the answer to the nearest tenth.

Find the derivative of $y=3 \sec x \tan x$ with respect to $x$ .

Find the integral of $e^{x}/x$ from 1 to 2, given $F(x)=\int_{0}^{x} e^{e^{t}} dt$ .

Find the value of the third derivative of $f(x)$ at $x=-1$ , given the degree 3 Taylor polynomial $P(x)=8-\frac{9}{10}(x+1)+\frac{8}{7}(x+1)^{2}-\frac{10}{3}(x+1)^{3}$ .

Evaluate the natural logarithm of $e^{7}$ .

Approximate the integral of $f(x) = 2 - x^2$ from 0 to 1 using Riemann sums. Lisa calculates the upper sum using the maximum function value, while Theo calculates the lower sum using the minimum function value. Compare the results for 5 and 500 rectangles.

Determine the intersection point of the tangents at points $A$ and $B$ on the graph of the function $f(x) = -\frac{1}{2}x^{2} - 2x$ , where $A(-1/1.5)$ and $B(1/2.5)$ .

Determine if $y=\frac{1}{4} e^{-3 x}$ represents exponential growth or decay. Identify the graph.

Find the derivative $\frac{d y}{d x}$ when $x=\sqrt{y}$ .

Find the critical points of the function $f(\theta) = 8 \sec \theta + 4 \tan \theta$ for $0 < \theta < 2\pi$ .

Find a function $f(x)$ with derivative $f'(x) = 2$ that passes through the point $(0, 3)$ .

Find the derivative of $f(x)=\sqrt{7x+4}$ using the limit definition. Then find the equation of the tangent line at $x=3$ .

Find the limit of the piecewise function $g(x)$ as $x$ approaches 2 from the right.

Find the values of $\ln 35.2$ and $\log \frac{4}{5}$ .

Find the term independent of $x$ in the expansion of $\left(\frac{1}{x}+x^{2}\right)^{9}$ .

Find the value of $s(r(4))$ where $r(x) = 2x - 2$ and $s(x) = -x^2$ .

Use leading coefficient test to determine if $y \to \infty$ or $y \to -\infty$ as $x \to -\infty$ for $y = 7x^5 - 6x^4$ .

Expand the expression $e^{x-\ln x}$ .

Determine if the graphs of the function family $f_a(x) = x^4 - ax^2$ have no local maximum for $a \leq 0$ .

Find the slope of the function $y = |x| + 1$ .

Find two functions $f$ and $g$ such that $(f \circ g)(x) = (6x+7)^7$ , where neither function is the identity function.

Compute $\Delta y$ and $dy$ for $y=\sqrt{x-1}$ , $x=2$ , $\Delta x=0.8$ . Round answers to three decimal places. Sketch a diagram showing $dx$ , $dy$ , and $\Delta y$ .

Find $b$ in $\ln(25b) = \square$ .

Find the values of $x$ where the function $f(x)$ is negative, based on the graph.

Find the exponential function $g(t) = ab^t$ given $g(10) = 95$ and $g(30) = 27$ . Round the solution to four decimal places.

Find the derivative of $f(x) = -\sqrt{x^3} - \frac{\sqrt{x}}{2}$ and evaluate it at $x = 3$ . Express the answer as a simplified radical fraction.

Find the limit of $(f(x+h) - f(x))/h$ as $h$ approaches 0, given $f(x) = x^2 - 8$ .

Determine the $x$ - and $y$ -coordinates of the inflection point of the function $f(x)=\frac{e^{x}}{1+e^{x}}$ .

Given functions $f(x) = 0.5x^2$ and $g(x) = 3x^3 + 1$ . Find the difference quotient for each interval: a) $[0; 2]$ , b) $[-1; 3]$ , c) $[-1; 1]$ , d) $[-2; -1]$ .

Determine if $E=mc^2$ is a power function. If so, state the power and constant of variation.

Describe the end behavior of the function $y=-|2x+2|+3$ . As $x \rightarrow-\infty, y \rightarrow+\infty$ ; As $x \rightarrow+\infty, y \rightarrow-\infty$ .

Given the function $V(x) = (x - 4)^2 + 6, x \geq 4$ , find the minimum value of the function.

Differentiate the function $y=\frac{1}{x+\frac{1}{x+1}}$ with respect to $x$ .

Finde kritische Stelle und Grenzwerte einer Funktion $f$ mit vertikaler Asymptote bei $x=1$ .

Evaluate $N(0)$ and $N(18)$ for $N(t) = \frac{12.80}{0.02 + 0.53^{t}}$ . Round answers to 2 decimal places.

Find the derivative of $f(x) = x - x^2$ using the limit definition of the derivative.

Find the absolute extrema and their $x$ -values for $f(x)=2x^3+8x^2+8x-2$ on $[-5,0]$ .

Find the critical point and determine the behavior of the function $y=x^{7/2}-4x^2$ for $x>0$ . Critical point: $c=$

Simplify $\frac{e^{x} + 7e^{6x}}{e^{x}}$ before finding its derivative.

Find the domain, vertical asymptote(s), and horizontal asymptote of $f(x)=\frac{1}{x+2}$ . Domain: $x\neq -2$ , Vertical Asymptote(s): $x=-2$ , Horizontal Asymptote: $y=0$ .

Find the $y$ -values of the function $f(x) = -3e^x$ to two decimal places when $x = -2, -1, 0, 1, 2$ .

Find the time when the drug concentration $C(t)=\frac{0.7 t}{t^{2}+49}$ is maximized, accurate to at least 2 decimal places.

Find the logarithmic function $f(t) = a + \log_b(t)$ passing through points $(4, -0.5)$ and $(16, 0)$ .

Determine the function resulting from horizontally compressing $y=f(x)$ by $1/3$ and vertically translating the graph 6 units down.

Determine the derivative of the function $f(x) = \frac{1}{2} x^{3}$ .

Solve for $x$ to two decimal places using the exponential equation $700=500(1.04)^{x}$ .