Calculus

Problem 2101

Find $x$ in $[0, 2\pi]$ where the tangent to $f(x)=5 \sin (x)+5 \cos (x)$ is horizontal.

See Solution

Problem 2102

Find $f^{\prime}(-\pi) + f^{\prime}\left(\frac{\pi}{2}\right)$ for the piecewise function $f(x)=\left\{\begin{array}{l}4 \sin (x), \text { if } x \leq 0 \\ 10 \cos (x), \text { if } x>0\end{array}\right.$

See Solution

Problem 2103

Find $f^{\prime}\left(\frac{\pi}{4}\right)+f^{\prime}\left(\frac{3 \pi}{4}\right)$ for the piecewise function $f(x)=\left\{\begin{array}{l}8 \tan (x), \text { if } 0 \leq x<\frac{\pi}{2} \\ 2 \cot (x), \text { if } \frac{\pi}{2} \leq x<\pi\end{array}\right.$

See Solution

Problem 2104

Find $\frac{d y}{d x}$ for $y=3 x^{3}+10 \sqrt{x}$ at $x=1$ . Choices: 4, 14, 19, 8, 29, None.

See Solution

Problem 2105

Find $\frac{d^{2} y}{d t^{2}}$ for $y=3 t^{3}+\frac{4}{t}$ at $t=-1$ . What is the result?

See Solution

Problem 2106

Find $f'(1)$ for $f(x)=A x^{4}-B x^{2}+C$ given $f''(1)=0$ and $f'''(-3)=18$ . Choices: 0, 6, 3, 2, $\frac{1}{4}$ , None.

See Solution

Problem 2107

Find $f'(1)$ for $f(x)=A x^{4}-B x^{2}+C$ given $f''(1)=0$ and $f'''(-3)=18$ . Options: -6, 6, 3, 2, $\frac{1}{4}$ , None.

See Solution

Problem 2108

Find the slope of the tangent line to $f(x)=x^{4}+5 x^{3}-x^{2}+8$ at $x=1$ .

See Solution

Problem 2109

Find the normal line equation for the function $f(x)=x^{3}+3x^{2}+2$ at the point $(-1,4)$ .

See Solution

Problem 2110

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

See Solution

Problem 2111

Find the values of $x$ in $[0, 2\pi]$ where the tangent line to $f(x)=4\sqrt{3}\sin(x)+4\cos(x)$ is horizontal.

See Solution

Problem 2112

Find $f^{\prime}(-2 \pi)+f^{\prime}\left(\frac{3 \pi}{2}\right)$ for the piecewise function $f(x)=\left\{\begin{array}{l}2 \sin (x), \text { if } x \leq 0 \\ 3 \cos (x), \text { if } x>0\end{array}\right.$

See Solution

Problem 2113

Find $f^{\prime}\left(\frac{\pi}{4}\right)+f^{\prime}\left(\frac{3 \pi}{4}\right)$ for the piecewise function: $f(x)=\begin{cases}4 \tan (x) & 0 \leq x<\frac{\pi}{2} \\ 8 \cot (x) & \frac{\pi}{2} \leq x<\pi\end{cases}$

See Solution

Problem 2114

Find $\frac{d y}{d x}$ for $y=4 x^{2}+2 \sqrt{x}$ at $x=1$ . What is the value?

See Solution

Problem 2115

Evaluate $\frac{d^{2} y}{d t^{2}}$ for $y=4 t^{3}+\frac{2}{t}$ at $t=-1$ . Options: $-26$ , $-20$ , 20, $-28$ , 8, None.

See Solution

Problem 2116

Find the value of $B$ for the function $f(x)=A x^{3}+B x^{2}+C$ given a horizontal tangent at $(-2, f(-2))$ and a normal slope of $\frac{-1}{15}$ at $(1, f(1))$ . Options: $0$ , $\frac{15}{4}$ , $\frac{25}{4}$ , $\frac{5}{2}$ , $\frac{15}{2}$ , None of the above.

See Solution

Problem 2117

Find $f'(1)$ for $f(x)=A x^{4}-B x^{2}+C$ given $f''(1)=0$ and $f'''(-4)=30$ . Choices: $\frac{15}{4}$ , $0-10$ , $\frac{5}{2}$ , $\frac{5}{16}$ , $\frac{15}{2}$ , None.

See Solution

Problem 2118

Find the derivative of $f(x)=4 x^{2}+11 x-2$ at $x=3$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}$ .

See Solution

Problem 2119

Find $f^{\prime}(-2 \pi) + f^{\prime}\left(\frac{3 \pi}{2}\right)$ for the piecewise function: $f(x)=\{2 \sin (x), x \leq 0; 3 \cos (x), x>0\}$ .

See Solution

Problem 2120

Calculate the average rate of change for $f(x)=3 x^{2}-2 x+5$ over the interval $[-2,6]$ .

See Solution

Problem 2121

Find $f^{\prime}(x)$ using the limit definition of the derivative for $f(x)=-4x^{2}-6x+7$ .

See Solution

Problem 2122

Find $f^{\prime}(x)$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{-2}{2 x-3}$ .

See Solution

Problem 2123

Find the expression for the derivative of $f(x)=4 x^{2}+11 x-2$ at $x=3$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}$ .

See Solution

Problem 2124

Find the tangent line equation for $f(x)=\frac{-2}{2x-3}$ at $x=2$ using results from problem $\neq 3$ .

See Solution

Problem 2125

Estimate the limit: $\lim _{x \rightarrow 0} \frac{x-8}{x-4}$ . If it doesn't exist, enter DNE.

See Solution

Problem 2126

Estimate the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-5}{x-2}$ . If no limit exists, write DNE.

See Solution

Problem 2127

Estimate the limit as $x$ approaches -1 for $\frac{x^{2}+2x+1}{x+1}$ . If none, enter DNE.

See Solution

Problem 2128

Estimate the limit numerically: $\lim _{x \rightarrow 9^{-}} \frac{\sqrt{9-x}}{9-x}$ . If no limit, enter DNE.

See Solution

Problem 2129

Estimate the limit numerically: $\lim _{x \rightarrow+\infty} \frac{7 x^{2}+6 x-6}{2 x^{2}-6 x}$ (If none, enter DNE.)

See Solution

Problem 2130

Graph the function $g(x)=\frac{4}{x^{2}-8}$ to check for continuity. List points of discontinuity or enter NONE.

See Solution

Problem 2131

Find the limit: $\lim _{x \rightarrow-2} \frac{x+4}{x}$ . What is the result?

See Solution

Problem 2132

Solve $\frac{3}{2} \int 4^{\frac{2 x}{3}} \frac{2}{3} d x=\left(\frac{3}{2}\right) \frac{4^{\frac{2 x}{3}}}{\ln 4}$ .

See Solution

Problem 2133

Find the terminal speed and time to reach the ground for a paratrooper with $\rho=0.075$ , $g=32 \mathrm{ft/s}^2$ , and starting at 10,000 ft.

See Solution

Problem 2134

Differentiate $x^{2}+1$ with respect to $x$ .

See Solution

Problem 2135

Cho hàm số liên tục $y=f(x)$ với bảng xét dấu đạo hàm. Hỏi hàm có bao nhiêu điểm cực trị? A. 1 B. 4 C. 2 D. 3

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Problem 2136

Differentiate $x^{2}+1$ with respect to $y$ .

See Solution

Problem 2137

Find the value of $\lim _{x \rightarrow 0^{-}} f(x)$ where $f(x)=\sqrt{-x}-1$ for $x<0$ , $-\sqrt{x}+1$ for $x>0$ , and $0$ for $x=0$ .

See Solution

Problem 2138

Simplify $\lim _{x \rightarrow 0} f(x)$ for the function $f(x)=\frac{1}{x-\frac{1}{x}}$ .

See Solution

Problem 2139

Find $\lim _{x \rightarrow-2} f(x)$ where $g(x)=\cos (\pi x)$ and $h(x)=-3 x^{4}-17 x^{3}-4 x^{2}+92 x+113$ for $-3<x<2$ .

See Solution

Problem 2140

A crossbow bolt is shot upward from $y=2$ at $t=0$ with initial velocity $v_{0}=45 \mathrm{~m/s}$ . When does it reach a max height of about $108.47 \mathrm{~m}$ ? Use $g=9.8 \mathrm{~m/s}^{2}$ .

See Solution

Problem 2141

Find the limit: $\lim _{x \rightarrow-2^{+}} \frac{x^{2}+2 x}{x^{3}+2 x^{2}-4 x-8}$ .

See Solution

Problem 2142

Check if the point M(-1, 1) is on the function $f(x) = 2 + 4x$ . Also, find the antiderivative $F(x)$ that passes through M.

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Problem 2143

A lime is thrown up at $5 \, \mathrm{m/s}$ . Find its max height, time to reach it, total air time, and impact velocity.

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Problem 2144

Найдите первообразную $F(x)$ для $f(x)=\sin \left(x-\frac{\pi}{4}\right)$ , проходящую через $M\left(\frac{3 \pi}{2} ; 2\right)$ .

See Solution

Problem 2145

Calculate the integral $\int\left(\frac{x^{-7}}{6}-1.25 x^{4}\right) d x$ .

See Solution

Problem 2146

Evaluate the integral: $\int\left(15 x^{24}-\frac{28}{\sqrt{6-7 x}}\right) d x$

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Problem 2147

What is the velocity of an object dropped for 3.0 s? Use $v = gt$ with $g \approx 9.81 \, \text{m/s}^2$ .

See Solution

Problem 2148

A ball is thrown up at $16 \mathrm{~m/s}$ . What is its velocity after $2.0 \mathrm{~s}$ ?

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Problem 2149

A stone is thrown down from a bridge at $5.6 \mathrm{~m/s}$ . What is its velocity after 3 seconds?

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Problem 2150

An arrow is shot up at $12.6 \mathrm{~m/s}$ . How long until it hits the ground?

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Problem 2151

Find the velocity function $v(t)$ for $s(t)=t^{2}-7t+6$ , and identify intervals for positive and negative motion.

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Problem 2152

Find the velocity function $v(t)$ for $s(t)=t^{3}-11 t^{2}+7 t-210$ , and intervals for positive/negative motion.

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Problem 2153

A 25 ft ladder leans against a wall. If the base moves away at 2 ft/sec, find the top's velocity at 7, 15, and 20 ft from the wall. Also, find the angle change rate when 7 ft from the wall.

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Problem 2154

Find the initial speed of water shooting to a height of $171 \mathrm{~m}$ .

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Problem 2155

Define $f(2)$ to make $f(x)=\frac{4-\sqrt{x^{2}+x+10}}{x-2}$ continuous at $x=2$ .

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Problem 2156

Determine the global min and max of the function $f(x)=-x^{4}+x^{3}+20x^{2}$ .

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Problem 2157

Find the point of inflection for $y=M(d)$ using $R(d)=\frac{1}{200}(-d^{4}+35 d^{3}-411 d^{2}+1845 d-2686.5)$ . Options: (A) 8.627 (B) 3.894, 13.728 (C) 5.911, 11.500 (D) 3.894, 8.627, 13.728.

See Solution

Problem 2158

An egg is dropped from a height of $61 \mathrm{~m}$ . Find the time to hit the ground and the impact speed.

See Solution

Problem 2159

Find $h^{\prime}(5)$ given that $h(x)=f(x) \cdot g(x)$ and $f^{\prime}(5)=10$ , $g^{\prime}(5)=10$ .

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Problem 2160

Find the local maxima and minima of the function $h$ , which peaks at (0,3) and decreases on both sides.

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Problem 2161

Find all local maximum values of the function $h$ that decreases from the point (0,3) and their corresponding x-values.

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Problem 2162

Differentiate $x^{2}+1$ with respect to $x$ . What is the result?

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Problem 2163

Find the derivative of $g(x)=\frac{5}{x}$ using the limit definition.

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Problem 2164

Find the derivative of $g(x)=\frac{5}{x}$ using the limit definition.

See Solution

Problem 2165

Identify the FALSE statement about the natural logarithmic function $y=\ln (x)$ from the options given.

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Problem 2166

Analyze the limits of $f(x) = -x^{8}$ as $x$ approaches $-\infty$ and $\infty$ . What do they approach?

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Problem 2167

Analyze the limits of $f(x)=-x^{3}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What are the results?

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Problem 2168

Evaluate the integrals: $\int \frac{dx}{\sin x \cos x}$ and $\int \frac{2dx}{\sin 2x}$ .

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Problem 2169

Find the value of $a$ where the tangent to $f(x)=2x^2+x+6$ is parallel to $y=5x+3$ .

See Solution

Problem 2170

Calculate the integral: $\int x(x+4)^{1 / 3} d x$

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Problem 2171

Find the derivative $f^{\prime}(4)$ for the function $f(x)=-\frac{\sqrt{x^{3}}}{3}+\frac{1}{x}$ .

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Problem 2172

Find the derivative $f^{\prime}(6)$ for the function $f(x)=\frac{4}{\sqrt{x}}+\frac{\sqrt{x^{3}}}{3}$ as a simplified fraction.

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Problem 2173

Find the second derivative $f^{\prime \prime}(x)$ of the function $f(x)=3x-\frac{1}{3}x^{4}+x^{2}-\frac{1}{4}x^{5}$ .

See Solution

Problem 2174

Find the derivative $f^{\prime}(4)$ for the function $f(x)=\frac{\sqrt{x^{3}}}{5}-\sqrt{x}$ .

See Solution

Problem 2175

Find the third derivative of the function $f(x)=-\frac{4}{3} x-\frac{1}{3} x^{5}+1$ .

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Problem 2176

Find the second derivative $f^{\prime \prime}(x)$ of the function $f(x)=-2 x^{-1}-\frac{1}{4} x^{4}-\frac{4}{5} x^{-4}$ .

See Solution

Problem 2177

Find the second derivative of the function $y=\frac{1}{6} x^{4}-\frac{2}{5}-\frac{2}{5} x$ .

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Problem 2178

Find a value for $a$ such that the solution to $\frac{d y}{d x}=(y^{2}-12y+27) \sin^2(\frac{2\pi y}{17})$ with $y(0)=a$ is non-constant and $\lim_{x \to \infty} y(x) = \frac{17}{2}$ .

See Solution

Problem 2179

Find the coefficients $c_{n}$ for the heat flow problem given $u(x, t)=\sum_{n=1}^{\infty} c_{n} e^{-n^{2} t} \sin (n x)$ and $u_{4}(x, 0)=f(x)$ .

See Solution

Problem 2180

Calculate the integral: $\int \sin (t) \sqrt{1+\cos (t)} \, dt$

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Problem 2181

Find the integral of $\frac{x^{2}}{1+x^{4}}$ with respect to $x$ .

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Problem 2182

Is it true or false that the derivative of a sum is the sum of the derivatives? Explain your answer. Options: A, B, C, D.

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Problem 2183

Is the statement true or false? The derivative of $f(x)=\frac{1}{x^{4}}$ is $f^{\prime}(x)=\frac{1}{4 x^{3}}$ . Choose A-E.

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Problem 2184

Find the derivative of the function $y=x^{3}-13 x^{2}+48 x+5$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 2185

Find the derivative of the function $f(u)=2 u^{0.8}-8 u^{2.6}$ . What is $f^{\prime}(u)$ ?

See Solution

Problem 2186

Find the derivative of the function $y=6 \sqrt{x}+8 x^{\frac{1}{5}}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 2187

Find the derivative of the function $y = 11 x^{2} - 9 x - 7 x^{-2}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 2188

Find the derivative of the function $y=\frac{7}{x^{4}}-\frac{9}{x}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 2189

Find the derivative of the function $p(x)=-28 x^{-\frac{2}{7}}+10 x^{-\frac{1}{5}}$ . What is $p^{\prime}(x)$ ?

See Solution

Problem 2190

Find the derivative of the function $y=\frac{-10}{\sqrt[3]{x}}$ . What is $\frac{d y}{d x}=?$

See Solution

Problem 2191

Find the derivative of $y=\frac{x^{5}+6}{x^{2}}$ after simplifying to $y=$ by dividing each term by $x^{2}$ .

See Solution

Problem 2192

Find the derivative of $y=\frac{x^{5}+6}{x^{2}}$ after simplifying it to $y=$ .

See Solution

Problem 2193

Find the derivative of $y=\frac{x^{5}+8}{x}$ after simplifying it. What is $y$ ?

See Solution

Problem 2194

Find the derivative of the function $y=\frac{x^{5}+11}{x}$ . What is $\frac{d y}{d x}=?$

See Solution

Problem 2195

Find the derivative of $h(x)=(x^{5}-2)^{3}$ . What is $h^{\prime}(x)=?$

See Solution

Problem 2196

Identify the correct type of derivative function $f^{\prime}(x)$ for a quadratic function $f(x)$ : Constant, Quadratic, Linear, or Cubic?

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Problem 2197

Explain how the slope and derivative of $f(x)$ at $x=a$ relate. Choose the correct answer from A, B, C, or D.

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Problem 2198

Find the derivative of the function: $D_{x}\left(9 x^{-\frac{1}{2}}+\frac{2}{x^{\frac{3}{2}}}\right)$ .

See Solution

Problem 2199

Find $f^{\prime}(-3)$ for $f(x)=\frac{x^{4}}{3}-5x$ . What is $f^{\prime}(-3)$ ?

See Solution

Problem 2200

Find the slope of the tangent line for $y=x^{4}-2x^{3}+8$ at $x=2$ and its equation. How to find the slope? A, B, C, or D?

See Solution

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Calculus

Problem 2101

Find xxx in [0,2π][0, 2\pi][0,2π] where the tangent to f(x)=5sin⁡(x)+5cos⁡(x)f(x)=5 \sin (x)+5 \cos (x)f(x)=5sin(x)+5cos(x) is horizontal.

Problem 2102

Problem 2103

Problem 2104

Find dydx\frac{d y}{d x}dxdy​ for y=3x3+10xy=3 x^{3}+10 \sqrt{x}y=3x3+10x​ at x=1x=1x=1. Choices: 4, 14, 19, 8, 29, None.

Problem 2105

Find d2ydt2\frac{d^{2} y}{d t^{2}}dt2d2y​ for y=3t3+4ty=3 t^{3}+\frac{4}{t}y=3t3+t4​ at t=−1t=-1t=−1. What is the result?

Problem 2106

Find f′(1)f'(1)f′(1) for f(x)=Ax4−Bx2+Cf(x)=A x^{4}-B x^{2}+Cf(x)=Ax4−Bx2+C given f′′(1)=0f''(1)=0f′′(1)=0 and f′′′(−3)=18f'''(-3)=18f′′′(−3)=18. Choices: 0, 6, 3, 2, 14\frac{1}{4}41​, None.

Problem 2107

Find f′(1)f'(1)f′(1) for f(x)=Ax4−Bx2+Cf(x)=A x^{4}-B x^{2}+Cf(x)=Ax4−Bx2+C given f′′(1)=0f''(1)=0f′′(1)=0 and f′′′(−3)=18f'''(-3)=18f′′′(−3)=18. Options: -6, 6, 3, 2, 14\frac{1}{4}41​, None.

Problem 2108

Find the slope of the tangent line to f(x)=x4+5x3−x2+8f(x)=x^{4}+5 x^{3}-x^{2}+8f(x)=x4+5x3−x2+8 at x=1x=1x=1.

Problem 2109

Find the normal line equation for the function f(x)=x3+3x2+2f(x)=x^{3}+3x^{2}+2f(x)=x3+3x2+2 at the point (−1,4)(-1,4)(−1,4).

Problem 2110

Find points where the tangent line of f(x)=13x3−52x2−x+1f(x)=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}-x+1f(x)=31​x3−25​x2−x+1 has slope -5.

Problem 2111

Find the values of xxx in [0,2π][0, 2\pi][0,2π] where the tangent line to f(x)=43sin⁡(x)+4cos⁡(x)f(x)=4\sqrt{3}\sin(x)+4\cos(x)f(x)=43​sin(x)+4cos(x) is horizontal.

Problem 2112

Problem 2113

Problem 2114

Find dydx\frac{d y}{d x}dxdy​ for y=4x2+2xy=4 x^{2}+2 \sqrt{x}y=4x2+2x​ at x=1x=1x=1. What is the value?

Problem 2115

Evaluate d2ydt2\frac{d^{2} y}{d t^{2}}dt2d2y​ for y=4t3+2ty=4 t^{3}+\frac{2}{t}y=4t3+t2​ at t=−1t=-1t=−1. Options: −26-26−26, −20-20−20, 20, −28-28−28, 8, None.

Problem 2116

Problem 2117

Find f′(1)f'(1)f′(1) for f(x)=Ax4−Bx2+Cf(x)=A x^{4}-B x^{2}+Cf(x)=Ax4−Bx2+C given f′′(1)=0f''(1)=0f′′(1)=0 and f′′′(−4)=30f'''(-4)=30f′′′(−4)=30. Choices: 154\frac{15}{4}415​, 0−100-100−10, 52\frac{5}{2}25​, 516\frac{5}{16}165​, 152\frac{15}{2}215​, None.

Problem 2118

Find the derivative of f(x)=4x2+11x−2f(x)=4 x^{2}+11 x-2f(x)=4x2+11x−2 at x=3x=3x=3 using f′(x)=lim⁡h→0f(3+h)−f(3)hf^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}f′(x)=limh→0​hf(3+h)−f(3)​.

Problem 2119

Find f′(−2π)+f′(3π2)f^{\prime}(-2 \pi) + f^{\prime}\left(\frac{3 \pi}{2}\right)f′(−2π)+f′(23π​) for the piecewise function: f(x)={2sin⁡(x),x≤0;3cos⁡(x),x>0}f(x)=\{2 \sin (x), x \leq 0; 3 \cos (x), x>0\}f(x)={2sin(x),x≤0;3cos(x),x>0}.

Problem 2120

Calculate the average rate of change for f(x)=3x2−2x+5f(x)=3 x^{2}-2 x+5f(x)=3x2−2x+5 over the interval [−2,6][-2,6][−2,6].

Problem 2121

Find f′(x)f^{\prime}(x)f′(x) using the limit definition of the derivative for f(x)=−4x2−6x+7f(x)=-4x^{2}-6x+7f(x)=−4x2−6x+7.

Problem 2122

Find f′(x)f^{\prime}(x)f′(x) using f′(x)=lim⁡h→0f(x+h)−f(x)hf^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​ for f(x)=−22x−3f(x)=\frac{-2}{2 x-3}f(x)=2x−3−2​.

Problem 2123

Find the expression for the derivative of f(x)=4x2+11x−2f(x)=4 x^{2}+11 x-2f(x)=4x2+11x−2 at x=3x=3x=3 using f′(x)=lim⁡h→0f(3+h)−f(3)hf^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}f′(x)=limh→0​hf(3+h)−f(3)​.

Problem 2124

Find the tangent line equation for f(x)=−22x−3f(x)=\frac{-2}{2x-3}f(x)=2x−3−2​ at x=2x=2x=2 using results from problem ≠3\neq 3=3.

Problem 2125

Estimate the limit: lim⁡x→0x−8x−4\lim _{x \rightarrow 0} \frac{x-8}{x-4}x→0lim​x−4x−8​. If it doesn't exist, enter DNE.

Problem 2126

Estimate the limit: lim⁡x→2x2−5x−2\lim _{x \rightarrow 2} \frac{x^{2}-5}{x-2}x→2lim​x−2x2−5​. If no limit exists, write DNE.

Problem 2127

Estimate the limit as xxx approaches -1 for x2+2x+1x+1\frac{x^{2}+2x+1}{x+1}x+1x2+2x+1​. If none, enter DNE.

Problem 2128

Estimate the limit numerically: lim⁡x→9−9−x9−x\lim _{x \rightarrow 9^{-}} \frac{\sqrt{9-x}}{9-x}x→9−lim​9−x9−x​​. If no limit, enter DNE.

Problem 2129

Estimate the limit numerically: lim⁡x→+∞7x2+6x−62x2−6x\lim _{x \rightarrow+\infty} \frac{7 x^{2}+6 x-6}{2 x^{2}-6 x}x→+∞lim​2x2−6x7x2+6x−6​ (If none, enter DNE.)

Problem 2130

Graph the function g(x)=4x2−8g(x)=\frac{4}{x^{2}-8}g(x)=x2−84​ to check for continuity. List points of discontinuity or enter NONE.

Problem 2131

Find the limit: lim⁡x→−2x+4x\lim _{x \rightarrow-2} \frac{x+4}{x}limx→−2​xx+4​. What is the result?

Problem 2132

Solve 32∫42x323dx=(32)42x3ln⁡4\frac{3}{2} \int 4^{\frac{2 x}{3}} \frac{2}{3} d x=\left(\frac{3}{2}\right) \frac{4^{\frac{2 x}{3}}}{\ln 4}23​∫432x​32​dx=(23​)ln4432x​​.

Problem 2133

Find the terminal speed and time to reach the ground for a paratrooper with ρ=0.075\rho=0.075ρ=0.075, g=32ft/s2g=32 \mathrm{ft/s}^2g=32ft/s2, and starting at 10,000 ft.

Problem 2134

Differentiate x2+1x^{2}+1x2+1 with respect to xxx.

Problem 2135

Cho hàm số liên tục y=f(x)y=f(x)y=f(x) với bảng xét dấu đạo hàm. Hỏi hàm có bao nhiêu điểm cực trị? A. 1 B. 4 C. 2 D. 3

Problem 2136

Differentiate x2+1x^{2}+1x2+1 with respect to yyy.

Problem 2137

Find the value of lim⁡x→0−f(x)\lim _{x \rightarrow 0^{-}} f(x)limx→0−​f(x) where f(x)=−x−1f(x)=\sqrt{-x}-1f(x)=−x​−1 for x<0x<0x<0, −x+1-\sqrt{x}+1−x​+1 for x>0x>0x>0, and 000 for x=0x=0x=0.

Problem 2138

Simplify lim⁡x→0f(x)\lim _{x \rightarrow 0} f(x)limx→0​f(x) for the function f(x)=1x−1xf(x)=\frac{1}{x-\frac{1}{x}}f(x)=x−x1​1​.

Problem 2139

Find lim⁡x→−2f(x)\lim _{x \rightarrow-2} f(x)limx→−2​f(x) where g(x)=cos⁡(πx)g(x)=\cos (\pi x)g(x)=cos(πx) and h(x)=−3x4−17x3−4x2+92x+113h(x)=-3 x^{4}-17 x^{3}-4 x^{2}+92 x+113h(x)=−3x4−17x3−4x2+92x+113 for −3<x<2-3<x<2−3<x<2.

Problem 2140

A crossbow bolt is shot upward from y=2y=2y=2 at t=0t=0t=0 with initial velocity v0=45 m/sv_{0}=45 \mathrm{~m/s}v0​=45 m/s. When does it reach a max height of about 108.47 m108.47 \mathrm{~m}108.47 m? Use g=9.8 m/s2g=9.8 \mathrm{~m/s}^{2}g=9.8 m/s2.

Problem 2141

Find the limit: lim⁡x→−2+x2+2xx3+2x2−4x−8\lim _{x \rightarrow-2^{+}} \frac{x^{2}+2 x}{x^{3}+2 x^{2}-4 x-8}limx→−2+​x3+2x2−4x−8x2+2x​.

Problem 2142

Check if the point M(-1, 1) is on the function f(x)=2+4xf(x) = 2 + 4xf(x)=2+4x. Also, find the antiderivative F(x)F(x)F(x) that passes through M.

Find $x$ in $[0, 2\pi]$ where the tangent to $f(x)=5 \sin (x)+5 \cos (x)$ is horizontal.

Find $\frac{d y}{d x}$ for $y=3 x^{3}+10 \sqrt{x}$ at $x=1$ . Choices: 4, 14, 19, 8, 29, None.

Find $\frac{d^{2} y}{d t^{2}}$ for $y=3 t^{3}+\frac{4}{t}$ at $t=-1$ . What is the result?

Find $f'(1)$ for $f(x)=A x^{4}-B x^{2}+C$ given $f''(1)=0$ and $f'''(-3)=18$ . Choices: 0, 6, 3, 2, $\frac{1}{4}$ , None.

Find $f'(1)$ for $f(x)=A x^{4}-B x^{2}+C$ given $f''(1)=0$ and $f'''(-3)=18$ . Options: -6, 6, 3, 2, $\frac{1}{4}$ , None.

Find the slope of the tangent line to $f(x)=x^{4}+5 x^{3}-x^{2}+8$ at $x=1$ .

Find the normal line equation for the function $f(x)=x^{3}+3x^{2}+2$ at the point $(-1,4)$ .

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

Find the values of $x$ in $[0, 2\pi]$ where the tangent line to $f(x)=4\sqrt{3}\sin(x)+4\cos(x)$ is horizontal.

Find $\frac{d y}{d x}$ for $y=4 x^{2}+2 \sqrt{x}$ at $x=1$ . What is the value?

Evaluate $\frac{d^{2} y}{d t^{2}}$ for $y=4 t^{3}+\frac{2}{t}$ at $t=-1$ . Options: $-26$ , $-20$ , 20, $-28$ , 8, None.

Find $f'(1)$ for $f(x)=A x^{4}-B x^{2}+C$ given $f''(1)=0$ and $f'''(-4)=30$ . Choices: $\frac{15}{4}$ , $0-10$ , $\frac{5}{2}$ , $\frac{5}{16}$ , $\frac{15}{2}$ , None.

Find the derivative of $f(x)=4 x^{2}+11 x-2$ at $x=3$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}$ .

Find $f^{\prime}(-2 \pi) + f^{\prime}\left(\frac{3 \pi}{2}\right)$ for the piecewise function: $f(x)=\{2 \sin (x), x \leq 0; 3 \cos (x), x>0\}$ .

Calculate the average rate of change for $f(x)=3 x^{2}-2 x+5$ over the interval $[-2,6]$ .

Find $f^{\prime}(x)$ using the limit definition of the derivative for $f(x)=-4x^{2}-6x+7$ .

Find $f^{\prime}(x)$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{-2}{2 x-3}$ .

Find the expression for the derivative of $f(x)=4 x^{2}+11 x-2$ at $x=3$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}$ .

Find the tangent line equation for $f(x)=\frac{-2}{2x-3}$ at $x=2$ using results from problem $\neq 3$ .

Estimate the limit: $\lim _{x \rightarrow 0} \frac{x-8}{x-4}$ . If it doesn't exist, enter DNE.

Estimate the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-5}{x-2}$ . If no limit exists, write DNE.

Estimate the limit as $x$ approaches -1 for $\frac{x^{2}+2x+1}{x+1}$ . If none, enter DNE.

Estimate the limit numerically: $\lim _{x \rightarrow 9^{-}} \frac{\sqrt{9-x}}{9-x}$ . If no limit, enter DNE.

Estimate the limit numerically: $\lim _{x \rightarrow+\infty} \frac{7 x^{2}+6 x-6}{2 x^{2}-6 x}$ (If none, enter DNE.)

Graph the function $g(x)=\frac{4}{x^{2}-8}$ to check for continuity. List points of discontinuity or enter NONE.

Find the limit: $\lim _{x \rightarrow-2} \frac{x+4}{x}$ . What is the result?

Solve $\frac{3}{2} \int 4^{\frac{2 x}{3}} \frac{2}{3} d x=\left(\frac{3}{2}\right) \frac{4^{\frac{2 x}{3}}}{\ln 4}$ .

Find the terminal speed and time to reach the ground for a paratrooper with $\rho=0.075$ , $g=32 \mathrm{ft/s}^2$ , and starting at 10,000 ft.

Differentiate $x^{2}+1$ with respect to $x$ .

Cho hàm số liên tục $y=f(x)$ với bảng xét dấu đạo hàm. Hỏi hàm có bao nhiêu điểm cực trị? A. 1 B. 4 C. 2 D. 3

Differentiate $x^{2}+1$ with respect to $y$ .

Find the value of $\lim _{x \rightarrow 0^{-}} f(x)$ where $f(x)=\sqrt{-x}-1$ for $x<0$ , $-\sqrt{x}+1$ for $x>0$ , and $0$ for $x=0$ .

Simplify $\lim _{x \rightarrow 0} f(x)$ for the function $f(x)=\frac{1}{x-\frac{1}{x}}$ .

Find $\lim _{x \rightarrow-2} f(x)$ where $g(x)=\cos (\pi x)$ and $h(x)=-3 x^{4}-17 x^{3}-4 x^{2}+92 x+113$ for $-3<x<2$ .

A crossbow bolt is shot upward from $y=2$ at $t=0$ with initial velocity $v_{0}=45 \mathrm{~m/s}$ . When does it reach a max height of about $108.47 \mathrm{~m}$ ? Use $g=9.8 \mathrm{~m/s}^{2}$ .

Find the limit: $\lim _{x \rightarrow-2^{+}} \frac{x^{2}+2 x}{x^{3}+2 x^{2}-4 x-8}$ .

Check if the point M(-1, 1) is on the function $f(x) = 2 + 4x$ . Also, find the antiderivative $F(x)$ that passes through M.

A lime is thrown up at $5 \, \mathrm{m/s}$ . Find its max height, time to reach it, total air time, and impact velocity.

Найдите первообразную $F(x)$ для $f(x)=\sin \left(x-\frac{\pi}{4}\right)$ , проходящую через $M\left(\frac{3 \pi}{2} ; 2\right)$ .

Calculate the integral $\int\left(\frac{x^{-7}}{6}-1.25 x^{4}\right) d x$ .

Evaluate the integral: $\int\left(15 x^{24}-\frac{28}{\sqrt{6-7 x}}\right) d x$

What is the velocity of an object dropped for 3.0 s? Use $v = gt$ with $g \approx 9.81 \, \text{m/s}^2$ .

A ball is thrown up at $16 \mathrm{~m/s}$ . What is its velocity after $2.0 \mathrm{~s}$ ?

A stone is thrown down from a bridge at $5.6 \mathrm{~m/s}$ . What is its velocity after 3 seconds?

An arrow is shot up at $12.6 \mathrm{~m/s}$ . How long until it hits the ground?

Find the velocity function $v(t)$ for $s(t)=t^{2}-7t+6$ , and identify intervals for positive and negative motion.

Find the velocity function $v(t)$ for $s(t)=t^{3}-11 t^{2}+7 t-210$ , and intervals for positive/negative motion.

Find the initial speed of water shooting to a height of $171 \mathrm{~m}$ .

Define $f(2)$ to make $f(x)=\frac{4-\sqrt{x^{2}+x+10}}{x-2}$ continuous at $x=2$ .

Determine the global min and max of the function $f(x)=-x^{4}+x^{3}+20x^{2}$ .

Find the point of inflection for $y=M(d)$ using $R(d)=\frac{1}{200}(-d^{4}+35 d^{3}-411 d^{2}+1845 d-2686.5)$ . Options: (A) 8.627 (B) 3.894, 13.728 (C) 5.911, 11.500 (D) 3.894, 8.627, 13.728.

An egg is dropped from a height of $61 \mathrm{~m}$ . Find the time to hit the ground and the impact speed.

Find $h^{\prime}(5)$ given that $h(x)=f(x) \cdot g(x)$ and $f^{\prime}(5)=10$ , $g^{\prime}(5)=10$ .

Find the local maxima and minima of the function $h$ , which peaks at (0,3) and decreases on both sides.

Find all local maximum values of the function $h$ that decreases from the point (0,3) and their corresponding x-values.

Differentiate $x^{2}+1$ with respect to $x$ . What is the result?

Find the derivative of $g(x)=\frac{5}{x}$ using the limit definition.

Find the derivative of $g(x)=\frac{5}{x}$ using the limit definition.

Identify the FALSE statement about the natural logarithmic function $y=\ln (x)$ from the options given.

Analyze the limits of $f(x) = -x^{8}$ as $x$ approaches $-\infty$ and $\infty$ . What do they approach?

Analyze the limits of $f(x)=-x^{3}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What are the results?

Evaluate the integrals: $\int \frac{dx}{\sin x \cos x}$ and $\int \frac{2dx}{\sin 2x}$ .

Find the value of $a$ where the tangent to $f(x)=2x^2+x+6$ is parallel to $y=5x+3$ .

Calculate the integral: $\int x(x+4)^{1 / 3} d x$

Find the derivative $f^{\prime}(4)$ for the function $f(x)=-\frac{\sqrt{x^{3}}}{3}+\frac{1}{x}$ .

Find the derivative $f^{\prime}(6)$ for the function $f(x)=\frac{4}{\sqrt{x}}+\frac{\sqrt{x^{3}}}{3}$ as a simplified fraction.

Find the second derivative $f^{\prime \prime}(x)$ of the function $f(x)=3x-\frac{1}{3}x^{4}+x^{2}-\frac{1}{4}x^{5}$ .

Find the derivative $f^{\prime}(4)$ for the function $f(x)=\frac{\sqrt{x^{3}}}{5}-\sqrt{x}$ .