Calculus

Problem 8901

Find the derivative of $f(x) = 30x^{\frac{1}{2}}$ .

See Solution

Problem 8902

Find the derivative of $f(x) = \frac{30}{\sqrt{x}} - x^{2}$ .

See Solution

Problem 8903

Find the derivative of the piecewise function $f(x) = \begin{cases} \frac{30}{\sqrt{x}} - x^2 \\ x^2 - 5x - 107 \end{cases}$ and evaluate it at $x=9$ .

See Solution

Problem 8904

The tangent line to $f$ at $(-2,3)$ is $y=-5x-1$ . Is $f^{\prime}(-2)=3$ true or false?

See Solution

Problem 8905

Find the rocket's climbing speed at $t = 6$ sec, given height $h = 4t^{2}$ ft. Answer in $\mathrm{ft} / \mathrm{sec}$ .

See Solution

Problem 8906

A boat is $2 \mathrm{mi}$ from shore, $11 \mathrm{mi}$ from a restaurant. Find landing point to minimize travel time and row speed.
a. Objective function for time $T$ is: $T=\frac{\sqrt{2^{2}+x^{2}}}{2}+\frac{11-x}{3}$ Interval: $[0,11]$ . Landing point: $11-\frac{4}{\sqrt{5}}$ miles.
b. Minimum rowing speed for direct route: $\square \mathrm{mi} / \mathrm{hr}$ .

See Solution

Problem 8907

Find the derivative $f'(1)$ for the piecewise function:
$f(x) = 3x + 2$ if $x < 1$ , $f(x) = 5$ if $x = 1$ , $f(x) = 8x + x^2 - 4$ if $x > 1$ .

See Solution

Problem 8908

Find the derivative $f'(1)$ for the piecewise function defined as:
$f(x) = 6x - 5$ if $x < 1$ , $f(x) = 1$ if $x = 1$ , $f(x) = 6x + x^2 - 6$ if $x > 1$ .

See Solution

Problem 8909

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

See Solution

Problem 8910

Find the rocket's speed at $\mathrm{t} = 4$ sec if height is $5 \mathrm{t}^{2} \mathrm{ft}$ . Answer: $100 \mathrm{ft} / \mathrm{sec}$ .

See Solution

Problem 8911

Find the tangent line equation at point (1,-3) for the curve $y=2 x^{3}+3 x-8$ . Choose from options a), b), c).

See Solution

Problem 8912

If $\cos(x) + \cos(y) = y$ , is it true that $\frac{dy}{dx} = -\frac{\sin(x)}{1+\sin(y)}$ ? True or False?

See Solution

Problem 8913

If $\cos (x)-\cos (y)=3 x$ , then $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sin (x)+3}{\sin (y)}$ . True or False?

See Solution

Problem 8914

Find the tangent line equation at point $(-2,1)$ for the curve $y=x^{3}+2 x^{2}+1$ . Choose the correct option.

See Solution

Problem 8915

Find $\frac{dy}{dx}$ for $x^{3} y^{2}=5$ . Is $\frac{dy}{dx}=-\frac{3 y}{2 x}$ true or false?

See Solution

Problem 8916

Check if the derivative of $\cos (x) + \cos (y) = y$ is $\frac{d y}{d x} = -\frac{\sin (x)}{1 + \sin (y)}$ .

See Solution

Problem 8917

Find the tangent line equation for $y=\frac{x^{3}}{2}$ at point (8,256) using the limit definition of slope. Show work.

See Solution

Problem 8918

If $x^{2} y^{4}=10$ , is it true that $\frac{\mathrm{d} y}{\mathrm{~d} y}=-\frac{2 x}{y}$ ? True or False?

See Solution

Problem 8919

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for $\sin(x+y) = \sin(x) + \cos(y)$ using implicit differentiation.

See Solution

Problem 8920

Find the tangent line equation for the curve $y=x^{2}+1$ at the point $(3,10)$ using the limit definition of slope.

See Solution

Problem 8921

Find the derivative of $x^{4} \ln (x)$ . Choose from: a) $4 x^{2}$ , b) $x^{3}(4 \ln (x)+1)$ , c) $x^{3}$ , d) $x^{4} e^{x}$ .

See Solution

Problem 8922

If $\sin (x)+\cos (y)=y$ , is it true that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\cos (x)}{1+\sin (y)}$ ? True or False?

See Solution

Problem 8923

Differentiate $x^{7} \ln (x)$ and choose the correct answer: a) $x^{6}(6 \ln (x)+1)$ b) $x^{7}(7 \ln (x)+1)$ c) $x^{6}(5 \ln (x)+1)$ d) $x^{6}(7 \ln (x)+1)$

See Solution

Problem 8924

Find the derivative of $\frac{2 x+3}{3 x-5}$ . What is the result? a) $2 / 3$ b) $\frac{12 x-1}{(3 x-5)^{2}}$ c) $-\frac{19}{(3 x-5)^{2}}$ d) $\frac{19}{(3 x-5)^{2}}$

See Solution

Problem 8925

Find the tangent line equation for $f(x)=x^{2}+2x$ at $x=5$ . Enter as $y=\text{type your answer} \quad x+\text{type your answer}$ .

See Solution

Problem 8926

Find $\frac{d y}{d x}$ for the equation $x^{2}+6 x y+y^{2}=-4$ .

See Solution

Problem 8927

Find the dimensions of a cylindrical can with volume $1024 \pi \mathrm{cm}^{3}$ that minimize the surface area.

See Solution

Problem 8928

If $\cos (x-y)=\cos (x)-\cos (y)$ , find $\frac{d y}{d x}$ : a) $\frac{\sin (x-y)+\sin (x)}{\sin (x-y)+\sin (y)}$ b) $-\frac{\sin (x-y)+\sin (x)}{\sin (x-y)+\sin (y)}$ .

See Solution

Problem 8929

Find the max and min values of $g(x, y)=3 x^{2}+8 y^{2}$ for $-5 \leq x \leq 5$ and $-5 \leq y \leq 7$ .

See Solution

Problem 8930

Determine the absolute max and min of $k(x, y)=-x^{2}-y^{2}+22x+22y$ with $0 \leq x \leq 12$ , $y \geq 0$ , and $x+y \leq 24$ .

See Solution

Problem 8931

Find $\lambda$ using Lagrange multipliers for the budget constraint $B=c_{1} x+c_{2} y$ in terms of $p_{x}, p_{y}, c_{1}, c_{2}$ .

See Solution

Problem 8932

Determine the max and min values of $k(x, y)=-x^{2}-y^{2}+22x+22y$ under $0 \leq x \leq 12$ , $y \geq 0$ , $x+y \leq 24$ .

See Solution

Problem 8933

Check if the derivative of $5x^2 + y^2 = 100$ with respect to $x$ is $-\frac{5x}{y}$ .

See Solution

Problem 8934

Find the slope of the tangent line to the curve $y=x^{2}+6x$ at the point where $x=-3$ using the limit definition.

See Solution

Problem 8935

If $\sin(y) - \sin(x) = 10x$ , is $\frac{dy}{dx} = \frac{\cos(x) - 10}{\cos(y)}$ true or false?

See Solution

Problem 8936

Dadas las funciones $f(u)=\sqrt{u}$ y $u(x)=x^{2}+2x$ , ¿cuál es la derivada $\frac{d f}{d x}$ correcta?

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Problem 8937

Verify if the derivative of $\cos (x) - \cos (y) = 3x$ is $\frac{d y}{d x} = \frac{\sin (x) + 3}{\sin (y)}$ . True or false?

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Problem 8938

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for $x^{4}-3 x^{2} y+3 y^{2}=7$ . Choose from options a) to d).

See Solution

Problem 8939

Find $\frac{d y}{d x}$ for the equation $x^{4}+6 x^{2} y+4 y=11$ . Choose from options a) to d).

See Solution

Problem 8940

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

See Solution

Problem 8941

Find the instantaneous velocity of a rock dropped from a 600 m cliff at $t=3$ s using the limit definition: $v_{\text{inst}}=\lim_{x \rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}$ .

See Solution

Problem 8942

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ if $\sin (x+y)=\cos (x)-\sin (y)$ . Choose a) b) c) or d).

See Solution

Problem 8943

Find the limits for the piecewise function $f(x)=\left\{-\frac{3}{x+6}, x<-6; 3x+22, x>-6\right\}$ . Calculate: $\lim _{x \rightarrow-6^{-}} f(x), \lim _{x \rightarrow-6^{+}} f(x), \lim _{x \rightarrow-6} f(x).$

See Solution

Problem 8944

Find the graph of $f(x)$ with these limits: $\lim _{x \rightarrow-2} f(x)=1$ , $\lim _{x \rightarrow 0^{-}} f(x)=-1$ , $\lim _{x \rightarrow 2^{+}} f(x)=\infty$ .

See Solution

Problem 8945

Find the second derivative of the function $y=4 x^{2}+10 x+2 x^{-3}$ . Show your work.

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Problem 8946

Given $f(5)=2$ and $\lim _{x \rightarrow 5}[f(x)+8 g(x)]=11$ , find: (a) $g(5)$ ; (b) $\lim _{x \rightarrow 5} g(x)$ .

See Solution

Problem 8947

Find the derivative of $y=8 x^{-2}+12 x^{3}-2 x$ and show your work.

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Problem 8948

Find the rock's falling speed at $t=3$ seconds using the limit definition of instantaneous velocity from $s=600-6.22 t^{2}$ .

See Solution

Problem 8949

Find the derivative of the function $w=z^{-6}-\frac{1}{z}$ . Show your work.

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Problem 8950

Evaluate the limits and value of the function $g(x)=\frac{5 x+9|x|}{3 x-8|x|}$ at $x=0$ . Write DNE if undefined.

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Problem 8951

Dibuja la gráfica de $f(x)$ con límites: $\lim _{x \rightarrow-2} f(x)=1$ , $\lim _{x \rightarrow 0^{+}} f(x)=-2$ , $\lim _{x \rightarrow 2} f(x)=2$ .

See Solution

Problem 8952

Dada la función $f(x)$ , analiza cómo serían las gráficas con los siguientes límites: $\lim _{x \rightarrow-2} f(x)=2$ , $\lim _{x \rightarrow 0^{-}} f(x)=0$ , $\lim _{x \rightarrow 2} f(x)=3$ , $\lim _{x \rightarrow-2} f(x)=1$ , $\lim _{x \rightarrow 0} f(x)=-1$ , $\lim _{x \rightarrow 2^{+}} f(x)=\infty$ , $\lim _{x \rightarrow 0^{+}} f(x)=-2$ .

See Solution

Problem 8953

Differentiate the function $f(x)=7 x^{8} \ln (x^{8}+1)$ with respect to $x$ .

See Solution

Problem 8954

Differentiate the function $h(t)=\frac{\ln t}{8+5 t^{3}}$ with respect to $t$ .

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Problem 8955

Find the derivative y' of the function $y=(3 x-4)(4 x^{3}-x^{2}+1)$ . Show your work.

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Problem 8956

Find the derivative of $y = t(\ln(4t))^{2}$ with respect to $t$ .

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Problem 8957

A sphere's radius $r$ increases at 5 in/min. Find volume change rates at $r=10$ in and $r=38$ in.

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Problem 8958

Find the derivative of $y=\frac{x^{2}+8 x+3}{\sqrt{x}}$ . Show your work.

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Problem 8959

A 25 ft ladder leans against a wall. The base moves away at 2 ft/sec. Find the top's velocity and area change rate at given distances.

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Problem 8960

Can the Mean Value Theorem apply to $f(x)=x^{2}$ on $[3,5]$ ? (Select all that apply.)

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Problem 8961

Find the derivative of $f(x)=7 x^{2} e^{x}$ . Show your work.

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Problem 8962

Find the limit as $h$ approaches $0$ : $\lim _{h \rightarrow 0} \frac{g(x+1) f(x+h)-g(x) f(x)}{h}$ .

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Problem 8963

Check if Rolle's Theorem applies to $f(x)=x^{2}-15x+14$ on the interval $[1,14]$ . Select all that apply.

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Problem 8964

Check if Rolle's Theorem applies to $f(x)=-x^{2}+4x$ on the interval $[0, 4]$ .

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Problem 8965

Differentiate the function $f(t)=\sqrt[5]{t}-\frac{1}{\sqrt[5]{t}}$ . Find $f^{\prime}(t)$ .

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Problem 8966

Find $a, b, c,$ and $d$ so that $f$ meets the Mean Value Theorem on [-1, 2] with $f(x)$ as defined.

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Problem 8967

Find the derivative of $y=\frac{8 e^{t}}{2 e^{t}+1}$ . Show your work.

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Problem 8968

Find all $c$ such that Rolle's theorem holds for $f(x) = 4 - 32x + 4x^{2}$ on $[3,5]$ .

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Problem 8969

A ball's height after $t$ seconds is given by $f(t)=-16 t^{2}+80 t+7$ .
(a) Show that $f(2)=f(3)$ . (b) Use Rolle's Theorem to find the velocity in $(2,3)$ and the time $t$ .

See Solution

Problem 8970

Find all $c$ in [0,25] that satisfy Rolle's Theorem for $f(x) = \sqrt{x} - \frac{1}{5}x$ .

See Solution

Problem 8971

Can the Mean Value Theorem be applied to $f(x)=|5x+5|$ on $[-3,3]$ ? Select all that apply.

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Problem 8972

Find the derivative of $f(x)=\frac{\sqrt{x}-2}{\sqrt{x}+2}$ and calculate $f^{\prime}(3)$ .

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Problem 8973

Find the derivative of $f(x)=\sqrt{2+2x}$ at $x=4$ using $f'(a)=\lim_{x \to a} \frac{f(x)-f(a)}{x-a}$ .

See Solution

Problem 8974

Find the minimum and maximum values of $f(8) - f(3)$ given $4 \leq f^{\prime}(x) \leq 5$ .

See Solution

Problem 8975

Find $f'(0)$ and $f'(2)$ for the function $f(x)=\cos\left(\frac{\pi x}{2}\right)$ .

See Solution

Problem 8976

For the function $y=2 x^{2}$ , find: (a) the average rate of change over $[3,6]$ and (b) the instantaneous rate at $x=3$ .

See Solution

Problem 8977

Find the tangent line equation for $y=\frac{8 x}{x^{2}+1}$ at point $(1,4)$ . Show your work.

See Solution

Problem 8978

Find the critical numbers of the function $f(x)=x^{6}-6 x^{5}$ . Enter answers as a comma-separated list.

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Problem 8979

Find the critical numbers of the function $g(t)=t \sqrt{6-t}$ for $t<\frac{13}{3}$ . Enter answers as a list.

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Problem 8980

Find the derivative value at the extremum: $f(x)=4-|x|$ , calculate $f^{\prime}(0)$ .

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Problem 8981

Define sequences: list, explicit, or recursive forms. Find limits or state "D" for divergence for given sequences.

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Problem 8982

Find the derivative of $y=g(x)=-x^{2}+9x+9$ , the slope at $x=2$ , $g(2)$ , and the tangent line equation.

See Solution

Problem 8983

Find the absolute extrema of $y=3 x^{2/3}-2 x$ on the interval $[-1,1]$ . Minimum $(x, y)=($ ) Maximum $(x, y)=($ )

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Problem 8984

Find the critical numbers of the function $(\sin (x))^{2}+\cos (x)$ for $0<x<2\pi$ . Enter answers as a list.

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Problem 8985

Find the absolute extrema of $f(x)=x^{3}-\frac{3}{2} x^{2}$ on the interval $[-2,4]$ . Minimum $(x, y)=(\ )$ Maximum $(x, y)=(\ )$

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Problem 8986

A ball dropped from a building has height $s(t)=256-16 t^{2}$ . Find the time to hit the ground and its impact velocity.

See Solution

Problem 8987

What is the minimum value of $f(6)$ given $f(4)=7$ and $f^{\prime}(x) \geq 2$ for $4 \leq x \leq 6$ ?

See Solution

Problem 8988

Find the limit: $\lim _{x \rightarrow \infty} 4 \cos x$ . If it doesn't exist, write DNE.

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Problem 8989

Find the speed and acceleration of the body at $t=2$ for the function $s=f(t)=7 t^{2}+2 t+8$ (0 ≤ $t$ ≤ 2).

See Solution

Problem 8990

Given the function $f(x)=\left\{\begin{aligned} 25-x^{2}, & x \leq 0 \\ -5 x, & x>0 \end{aligned}\right.$ , find critical numbers, intervals of increase/decrease, and relative extrema.

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Problem 8991

Find the limit: $\lim _{x \rightarrow \infty} \frac{3 x-8}{4 x+5}$ .

See Solution

Problem 8992

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(\sqrt{81 x^{2}+x}-9 x\right)$ .

See Solution

Problem 8993

Find the limit: $\lim _{x \rightarrow \infty} 7 \cos \frac{2}{3 x}$ .

See Solution

Problem 8994

Find the limit as $x$ approaches $-\infty$ for the expression $5x + \sqrt{25x^2 - x}$ .

See Solution

Problem 8995

Given the function $f(x)=9x+\frac{7}{x}$ , find:
(a) Critical numbers: $x=$
(b) Intervals where $f$ is increasing and decreasing.
Increasing: $(-\infty,-\frac{1}{3} \sqrt{7}), (-\frac{1}{3} \sqrt{7}, 0), (0, \frac{1}{3} \sqrt{7}), (\frac{1}{3} \sqrt{7}, \infty)$
Decreasing: $(-\infty,-\frac{1}{3} \sqrt{7}), (-\frac{1}{3} \sqrt{7}, 0), (0, \frac{1}{3} \sqrt{7}), (\frac{1}{3} \sqrt{7}, \infty)$
(c) Use the First Derivative Test for relative extremum: max $(x, y)=($ , min $(x, y)=1$

See Solution

Problem 8996

Find the limit: $\lim _{x \rightarrow \infty} \frac{9 x^{3}+1}{27 x^{3}-5 x^{2}+1}$ .

See Solution

Problem 8997

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x}{\sqrt{x^{2}-x}}$ .

See Solution

Problem 8998

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{x^{9}+2}{x^{9}-8}$ , (b) $\lim _{x \rightarrow \infty} \frac{x^{9}+2}{x^{10}-8}$ , (c) $\lim _{x \rightarrow \infty} \frac{x^{9}+2}{x^{8}-8}$ .

See Solution

Problem 8999

Find the next approximation $x_{1}$ using Newton's method for the root of $f(x)=0$ , starting with $x_{0}=2$ .

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Problem 9000

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{2 x^{3 / 2}}{5 x^{2}+8}=$ (b) $\lim _{x \rightarrow \infty} \frac{2 x^{3 / 2}}{5 x^{3 / 2}+8}=$ (c) $\lim _{x \rightarrow \infty} \frac{2 x^{3 / 2}}{5 \sqrt{x}+8}=$

See Solution

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Calculus

Problem 8901

Find the derivative of f(x)=30x12f(x) = 30x^{\frac{1}{2}}f(x)=30x21​.

Problem 8902

Find the derivative of f(x)=30x−x2f(x) = \frac{30}{\sqrt{x}} - x^{2}f(x)=x​30​−x2.

Problem 8903

Find the derivative of the piecewise function f(x)={30x−x2x2−5x−107f(x) = \begin{cases} \frac{30}{\sqrt{x}} - x^2 \\ x^2 - 5x - 107 \end{cases}f(x)={x​30​−x2x2−5x−107​ and evaluate it at x=9x=9x=9.

Problem 8904

The tangent line to fff at (−2,3)(-2,3)(−2,3) is y=−5x−1y=-5x-1y=−5x−1. Is f′(−2)=3f^{\prime}(-2)=3f′(−2)=3 true or false?

Problem 8905

Find the rocket's climbing speed at t=6t = 6t=6 sec, given height h=4t2h = 4t^{2}h=4t2 ft. Answer in ft/sec\mathrm{ft} / \mathrm{sec}ft/sec.

Problem 8906

Problem 8907

Find the derivative f′(1)f'(1)f′(1) for the piecewise function: f(x)=3x+2f(x) = 3x + 2f(x)=3x+2 if x<1x < 1x<1, f(x)=5f(x) = 5f(x)=5 if x=1x = 1x=1, f(x)=8x+x2−4f(x) = 8x + x^2 - 4f(x)=8x+x2−4 if x>1x > 1x>1.

Problem 8908

Find the derivative f′(1)f'(1)f′(1) for the piecewise function defined as: f(x)=6x−5f(x) = 6x - 5f(x)=6x−5 if x<1x < 1x<1, f(x)=1f(x) = 1f(x)=1 if x=1x = 1x=1, f(x)=6x+x2−6f(x) = 6x + x^2 - 6f(x)=6x+x2−6 if x>1x > 1x>1.

Problem 8909

Find the derivative of 1x2\frac{1}{x^{2}}x21​ with respect to xxx.

Problem 8910

Find the rocket's speed at t=4\mathrm{t} = 4t=4 sec if height is 5t2ft5 \mathrm{t}^{2} \mathrm{ft}5t2ft. Answer: 100ft/sec100 \mathrm{ft} / \mathrm{sec}100ft/sec.

Problem 8911

Find the tangent line equation at point (1,-3) for the curve y=2x3+3x−8y=2 x^{3}+3 x-8y=2x3+3x−8. Choose from options a), b), c).

Problem 8912

If cos⁡(x)+cos⁡(y)=y\cos(x) + \cos(y) = ycos(x)+cos(y)=y, is it true that dydx=−sin⁡(x)1+sin⁡(y)\frac{dy}{dx} = -\frac{\sin(x)}{1+\sin(y)}dxdy​=−1+sin(y)sin(x)​? True or False?

Problem 8913

If cos⁡(x)−cos⁡(y)=3x\cos (x)-\cos (y)=3 xcos(x)−cos(y)=3x, then dy dx=sin⁡(x)+3sin⁡(y)\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sin (x)+3}{\sin (y)} dxdy​=sin(y)sin(x)+3​. True or False?

Problem 8914

Find the tangent line equation at point (−2,1)(-2,1)(−2,1) for the curve y=x3+2x2+1y=x^{3}+2 x^{2}+1y=x3+2x2+1. Choose the correct option.

Problem 8915

Find dydx\frac{dy}{dx}dxdy​ for x3y2=5x^{3} y^{2}=5x3y2=5. Is dydx=−3y2x\frac{dy}{dx}=-\frac{3 y}{2 x}dxdy​=−2x3y​ true or false?

Problem 8916

Check if the derivative of cos⁡(x)+cos⁡(y)=y\cos (x) + \cos (y) = ycos(x)+cos(y)=y is dydx=−sin⁡(x)1+sin⁡(y)\frac{d y}{d x} = -\frac{\sin (x)}{1 + \sin (y)}dxdy​=−1+sin(y)sin(x)​.

Problem 8917

Find the tangent line equation for y=x32y=\frac{x^{3}}{2}y=2x3​ at point (8,256) using the limit definition of slope. Show work.

Problem 8918

If x2y4=10x^{2} y^{4}=10x2y4=10, is it true that dy dy=−2xy\frac{\mathrm{d} y}{\mathrm{~d} y}=-\frac{2 x}{y} dydy​=−y2x​? True or False?

Problem 8919

Find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} dxdy​ for sin⁡(x+y)=sin⁡(x)+cos⁡(y)\sin(x+y) = \sin(x) + \cos(y)sin(x+y)=sin(x)+cos(y) using implicit differentiation.

Problem 8920

Find the tangent line equation for the curve y=x2+1y=x^{2}+1y=x2+1 at the point (3,10)(3,10)(3,10) using the limit definition of slope.

Problem 8921

Find the derivative of x4ln⁡(x)x^{4} \ln (x)x4ln(x). Choose from: a) 4x24 x^{2}4x2, b) x3(4ln⁡(x)+1)x^{3}(4 \ln (x)+1)x3(4ln(x)+1), c) x3x^{3}x3, d) x4exx^{4} e^{x}x4ex.

Problem 8922

If sin⁡(x)+cos⁡(y)=y\sin (x)+\cos (y)=ysin(x)+cos(y)=y, is it true that dy dx=cos⁡(x)1+sin⁡(y)\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\cos (x)}{1+\sin (y)} dxdy​=1+sin(y)cos(x)​? True or False?

Problem 8923

Differentiate x7ln⁡(x) x^{7} \ln (x) x7ln(x) and choose the correct answer: a) x6(6ln⁡(x)+1) x^{6}(6 \ln (x)+1) x6(6ln(x)+1) b) x7(7ln⁡(x)+1) x^{7}(7 \ln (x)+1) x7(7ln(x)+1) c) x6(5ln⁡(x)+1) x^{6}(5 \ln (x)+1) x6(5ln(x)+1) d) x6(7ln⁡(x)+1) x^{6}(7 \ln (x)+1) x6(7ln(x)+1)

Problem 8924

Find the derivative of 2x+33x−5\frac{2 x+3}{3 x-5}3x−52x+3​. What is the result? a) 2/32 / 32/3 b) 12x−1(3x−5)2\frac{12 x-1}{(3 x-5)^{2}}(3x−5)212x−1​ c) −19(3x−5)2-\frac{19}{(3 x-5)^{2}}−(3x−5)219​ d) 19(3x−5)2\frac{19}{(3 x-5)^{2}}(3x−5)219​

Problem 8925

Find the tangent line equation for f(x)=x2+2xf(x)=x^{2}+2xf(x)=x2+2x at x=5x=5x=5. Enter as y=type your answerx+type your answery=\text{type your answer} \quad x+\text{type your answer}y=type your answerx+type your answer.

Problem 8926

Find dydx\frac{d y}{d x}dxdy​ for the equation x2+6xy+y2=−4x^{2}+6 x y+y^{2}=-4x2+6xy+y2=−4.

Problem 8927

Find the dimensions of a cylindrical can with volume 1024πcm31024 \pi \mathrm{cm}^{3}1024πcm3 that minimize the surface area.

Problem 8928

Problem 8929

Find the max and min values of g(x,y)=3x2+8y2g(x, y)=3 x^{2}+8 y^{2}g(x,y)=3x2+8y2 for −5≤x≤5-5 \leq x \leq 5−5≤x≤5 and −5≤y≤7-5 \leq y \leq 7−5≤y≤7.

Problem 8930

Determine the absolute max and min of k(x,y)=−x2−y2+22x+22yk(x, y)=-x^{2}-y^{2}+22x+22yk(x,y)=−x2−y2+22x+22y with 0≤x≤120 \leq x \leq 120≤x≤12, y≥0y \geq 0y≥0, and x+y≤24x+y \leq 24x+y≤24.

Problem 8931

Find λ\lambdaλ using Lagrange multipliers for the budget constraint B=c1x+c2yB=c_{1} x+c_{2} yB=c1​x+c2​y in terms of px,py,c1,c2p_{x}, p_{y}, c_{1}, c_{2}px​,py​,c1​,c2​.

Problem 8932

Determine the max and min values of k(x,y)=−x2−y2+22x+22yk(x, y)=-x^{2}-y^{2}+22x+22yk(x,y)=−x2−y2+22x+22y under 0≤x≤120 \leq x \leq 120≤x≤12, y≥0y \geq 0y≥0, x+y≤24x+y \leq 24x+y≤24.

Problem 8933

Check if the derivative of 5x2+y2=1005x^2 + y^2 = 1005x2+y2=100 with respect to xxx is −5xy-\frac{5x}{y}−y5x​.

Problem 8934

Find the slope of the tangent line to the curve y=x2+6xy=x^{2}+6xy=x2+6x at the point where x=−3x=-3x=−3 using the limit definition.

Problem 8935

If sin⁡(y)−sin⁡(x)=10x\sin(y) - \sin(x) = 10xsin(y)−sin(x)=10x, is dydx=cos⁡(x)−10cos⁡(y)\frac{dy}{dx} = \frac{\cos(x) - 10}{\cos(y)}dxdy​=cos(y)cos(x)−10​ true or false?

Problem 8936

Dadas las funciones f(u)=uf(u)=\sqrt{u}f(u)=u​ y u(x)=x2+2xu(x)=x^{2}+2xu(x)=x2+2x, ¿cuál es la derivada dfdx\frac{d f}{d x}dxdf​ correcta?

Problem 8937

Verify if the derivative of cos⁡(x)−cos⁡(y)=3x\cos (x) - \cos (y) = 3xcos(x)−cos(y)=3x is dydx=sin⁡(x)+3sin⁡(y)\frac{d y}{d x} = \frac{\sin (x) + 3}{\sin (y)}dxdy​=sin(y)sin(x)+3​. True or false?

Problem 8938

Find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} dxdy​ for x4−3x2y+3y2=7x^{4}-3 x^{2} y+3 y^{2}=7x4−3x2y+3y2=7. Choose from options a) to d).

Problem 8939

Find dydx\frac{d y}{d x}dxdy​ for the equation x4+6x2y+4y=11x^{4}+6 x^{2} y+4 y=11x4+6x2y+4y=11. Choose from options a) to d).

Problem 8940

Find the limit: lim⁡x→−∞−4e−x+2exe−x+3ex\lim _{x \rightarrow-\infty} \frac{-4 e^{-x}+2 e^{x}}{e^{-x}+3 e^{x}}limx→−∞​e−x+3ex−4e−x+2ex​.

Problem 8941

Find the derivative of $f(x) = 30x^{\frac{1}{2}}$ .

Find the derivative of $f(x) = \frac{30}{\sqrt{x}} - x^{2}$ .

Find the derivative of the piecewise function $f(x) = \begin{cases} \frac{30}{\sqrt{x}} - x^2 \\ x^2 - 5x - 107 \end{cases}$ and evaluate it at $x=9$ .

The tangent line to $f$ at $(-2,3)$ is $y=-5x-1$ . Is $f^{\prime}(-2)=3$ true or false?

Find the rocket's climbing speed at $t = 6$ sec, given height $h = 4t^{2}$ ft. Answer in $\mathrm{ft} / \mathrm{sec}$ .

Find the derivative $f'(1)$ for the piecewise function:
$f(x) = 3x + 2$ if $x < 1$ , $f(x) = 5$ if $x = 1$ , $f(x) = 8x + x^2 - 4$ if $x > 1$ .

Find the derivative $f'(1)$ for the piecewise function defined as:
$f(x) = 6x - 5$ if $x < 1$ , $f(x) = 1$ if $x = 1$ , $f(x) = 6x + x^2 - 6$ if $x > 1$ .

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

Find the rocket's speed at $\mathrm{t} = 4$ sec if height is $5 \mathrm{t}^{2} \mathrm{ft}$ . Answer: $100 \mathrm{ft} / \mathrm{sec}$ .

Find the tangent line equation at point (1,-3) for the curve $y=2 x^{3}+3 x-8$ . Choose from options a), b), c).

If $\cos(x) + \cos(y) = y$ , is it true that $\frac{dy}{dx} = -\frac{\sin(x)}{1+\sin(y)}$ ? True or False?

If $\cos (x)-\cos (y)=3 x$ , then $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sin (x)+3}{\sin (y)}$ . True or False?

Find the tangent line equation at point $(-2,1)$ for the curve $y=x^{3}+2 x^{2}+1$ . Choose the correct option.

Find $\frac{dy}{dx}$ for $x^{3} y^{2}=5$ . Is $\frac{dy}{dx}=-\frac{3 y}{2 x}$ true or false?

Check if the derivative of $\cos (x) + \cos (y) = y$ is $\frac{d y}{d x} = -\frac{\sin (x)}{1 + \sin (y)}$ .

Find the tangent line equation for $y=\frac{x^{3}}{2}$ at point (8,256) using the limit definition of slope. Show work.

If $x^{2} y^{4}=10$ , is it true that $\frac{\mathrm{d} y}{\mathrm{~d} y}=-\frac{2 x}{y}$ ? True or False?

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for $\sin(x+y) = \sin(x) + \cos(y)$ using implicit differentiation.

Find the tangent line equation for the curve $y=x^{2}+1$ at the point $(3,10)$ using the limit definition of slope.

Find the derivative of $x^{4} \ln (x)$ . Choose from: a) $4 x^{2}$ , b) $x^{3}(4 \ln (x)+1)$ , c) $x^{3}$ , d) $x^{4} e^{x}$ .

If $\sin (x)+\cos (y)=y$ , is it true that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\cos (x)}{1+\sin (y)}$ ? True or False?

Differentiate $x^{7} \ln (x)$ and choose the correct answer: a) $x^{6}(6 \ln (x)+1)$ b) $x^{7}(7 \ln (x)+1)$ c) $x^{6}(5 \ln (x)+1)$ d) $x^{6}(7 \ln (x)+1)$

Find the derivative of $\frac{2 x+3}{3 x-5}$ . What is the result? a) $2 / 3$ b) $\frac{12 x-1}{(3 x-5)^{2}}$ c) $-\frac{19}{(3 x-5)^{2}}$ d) $\frac{19}{(3 x-5)^{2}}$

Find the tangent line equation for $f(x)=x^{2}+2x$ at $x=5$ . Enter as $y=\text{type your answer} \quad x+\text{type your answer}$ .

Find $\frac{d y}{d x}$ for the equation $x^{2}+6 x y+y^{2}=-4$ .

Find the dimensions of a cylindrical can with volume $1024 \pi \mathrm{cm}^{3}$ that minimize the surface area.

Find the max and min values of $g(x, y)=3 x^{2}+8 y^{2}$ for $-5 \leq x \leq 5$ and $-5 \leq y \leq 7$ .

Determine the absolute max and min of $k(x, y)=-x^{2}-y^{2}+22x+22y$ with $0 \leq x \leq 12$ , $y \geq 0$ , and $x+y \leq 24$ .

Find $\lambda$ using Lagrange multipliers for the budget constraint $B=c_{1} x+c_{2} y$ in terms of $p_{x}, p_{y}, c_{1}, c_{2}$ .

Determine the max and min values of $k(x, y)=-x^{2}-y^{2}+22x+22y$ under $0 \leq x \leq 12$ , $y \geq 0$ , $x+y \leq 24$ .

Check if the derivative of $5x^2 + y^2 = 100$ with respect to $x$ is $-\frac{5x}{y}$ .

Find the slope of the tangent line to the curve $y=x^{2}+6x$ at the point where $x=-3$ using the limit definition.

If $\sin(y) - \sin(x) = 10x$ , is $\frac{dy}{dx} = \frac{\cos(x) - 10}{\cos(y)}$ true or false?

Dadas las funciones $f(u)=\sqrt{u}$ y $u(x)=x^{2}+2x$ , ¿cuál es la derivada $\frac{d f}{d x}$ correcta?

Verify if the derivative of $\cos (x) - \cos (y) = 3x$ is $\frac{d y}{d x} = \frac{\sin (x) + 3}{\sin (y)}$ . True or false?

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for $x^{4}-3 x^{2} y+3 y^{2}=7$ . Choose from options a) to d).

Find $\frac{d y}{d x}$ for the equation $x^{4}+6 x^{2} y+4 y=11$ . Choose from options a) to d).

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

Find the instantaneous velocity of a rock dropped from a 600 m cliff at $t=3$ s using the limit definition: $v_{\text{inst}}=\lim_{x \rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}$ .

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ if $\sin (x+y)=\cos (x)-\sin (y)$ . Choose a) b) c) or d).

Find the graph of $f(x)$ with these limits: $\lim _{x \rightarrow-2} f(x)=1$ , $\lim _{x \rightarrow 0^{-}} f(x)=-1$ , $\lim _{x \rightarrow 2^{+}} f(x)=\infty$ .

Find the second derivative of the function $y=4 x^{2}+10 x+2 x^{-3}$ . Show your work.

Given $f(5)=2$ and $\lim _{x \rightarrow 5}[f(x)+8 g(x)]=11$ , find: (a) $g(5)$ ; (b) $\lim _{x \rightarrow 5} g(x)$ .

Find the derivative of $y=8 x^{-2}+12 x^{3}-2 x$ and show your work.

Find the rock's falling speed at $t=3$ seconds using the limit definition of instantaneous velocity from $s=600-6.22 t^{2}$ .

Find the derivative of the function $w=z^{-6}-\frac{1}{z}$ . Show your work.

Evaluate the limits and value of the function $g(x)=\frac{5 x+9|x|}{3 x-8|x|}$ at $x=0$ . Write DNE if undefined.

Dibuja la gráfica de $f(x)$ con límites: $\lim _{x \rightarrow-2} f(x)=1$ , $\lim _{x \rightarrow 0^{+}} f(x)=-2$ , $\lim _{x \rightarrow 2} f(x)=2$ .

Differentiate the function $f(x)=7 x^{8} \ln (x^{8}+1)$ with respect to $x$ .

Differentiate the function $h(t)=\frac{\ln t}{8+5 t^{3}}$ with respect to $t$ .

Find the derivative y' of the function $y=(3 x-4)(4 x^{3}-x^{2}+1)$ . Show your work.

Find the derivative of $y = t(\ln(4t))^{2}$ with respect to $t$ .

A sphere's radius $r$ increases at 5 in/min. Find volume change rates at $r=10$ in and $r=38$ in.

Find the derivative of $y=\frac{x^{2}+8 x+3}{\sqrt{x}}$ . Show your work.

Can the Mean Value Theorem apply to $f(x)=x^{2}$ on $[3,5]$ ? (Select all that apply.)

Find the derivative of $f(x)=7 x^{2} e^{x}$ . Show your work.

Find the limit as $h$ approaches $0$ : $\lim _{h \rightarrow 0} \frac{g(x+1) f(x+h)-g(x) f(x)}{h}$ .

Check if Rolle's Theorem applies to $f(x)=x^{2}-15x+14$ on the interval $[1,14]$ . Select all that apply.

Check if Rolle's Theorem applies to $f(x)=-x^{2}+4x$ on the interval $[0, 4]$ .

Differentiate the function $f(t)=\sqrt[5]{t}-\frac{1}{\sqrt[5]{t}}$ . Find $f^{\prime}(t)$ .

Find $a, b, c,$ and $d$ so that $f$ meets the Mean Value Theorem on [-1, 2] with $f(x)$ as defined.

Find the derivative of $y=\frac{8 e^{t}}{2 e^{t}+1}$ . Show your work.

Find all $c$ such that Rolle's theorem holds for $f(x) = 4 - 32x + 4x^{2}$ on $[3,5]$ .

A ball's height after $t$ seconds is given by $f(t)=-16 t^{2}+80 t+7$ .
(a) Show that $f(2)=f(3)$ . (b) Use Rolle's Theorem to find the velocity in $(2,3)$ and the time $t$ .

Find all $c$ in [0,25] that satisfy Rolle's Theorem for $f(x) = \sqrt{x} - \frac{1}{5}x$ .

Can the Mean Value Theorem be applied to $f(x)=|5x+5|$ on $[-3,3]$ ? Select all that apply.

Find the derivative of $f(x)=\frac{\sqrt{x}-2}{\sqrt{x}+2}$ and calculate $f^{\prime}(3)$ .

Find the derivative of $f(x)=\sqrt{2+2x}$ at $x=4$ using $f'(a)=\lim_{x \to a} \frac{f(x)-f(a)}{x-a}$ .

Find the minimum and maximum values of $f(8) - f(3)$ given $4 \leq f^{\prime}(x) \leq 5$ .