Recent questions tagged goclasses-calculus-&-optimization-practice-questions

Find the length of the curve $x=\cos t,\ y=t+\sin t,\ 0\le t\le \pi$.

Find $\dfrac{dy}{dx}$ where $$y=\int_{x^2}^{\sin x}\sqrt{1+t^4}dt$$$\cos x\sqrt{1+\sin^4x}-2x\sqrt{1+x^8}$ $\cos x\sqrt{1+\sin^4x}+2x\sqrt{1+x^8}$ $\sqrt{1+\sin^4x}-\sqrt...

Find $f(x)$ if $f'(x)=6x^2+\sec^2 x$ and $f(0)=4$.$f(x)=2x^3+\tan x+4$ $f(x)=2x^3-\tan x+4$ $f(x)=6x^3+\tan x+4$ $f(x)=2x^3+\tan x-4$

Find the area of the region bounded by the lines $x=-1$, $x=2$, $y=0$, and the curve $y=x^3-x^2-2x$.$\frac{5}{12}$ $\frac{8}{3}$ $\frac{37}{12}$ $\frac{11}{4}$

The volume of the solid generated by revolving the region bounded by the curves $y=x^2$ and $y=x+2$ about the $x$-axis is:$\frac{72}{5}$ $\frac{72\pi}{5}$ $\frac{36\pi}{5...

The length of the curve $y=\frac{1}{3}\left((x-1)^2+2\right)^{3/2}$ for $-1\leq x\leq2$ is:$5$ $6$ $\dfrac{20}{3}$ $8$

The area of the region between the curves $y=x^2-2x$ and $y=2x-x^2$ for $x\in[-1,3]$ is:$4$ $\frac{16}{3}$ $\frac{20}{3}$ $8$

The point of inflection of the function $y=e^x(\sin x-\cos x)$ in $[0,\pi]$ is:$x=\dfrac{\pi}{4}$ $x=\dfrac{3\pi}{4}$ $x=\dfrac{\pi}{4},\dfrac{3\pi}{4}$ No point of infle...

The function $$f(x)=\begin{cases}x\sin\left(\frac{1}{x}\right),&x\neq0\\\\0,&x=0\end{cases}$$ at $x=0$ is:Continuous and differentiable Continuous but not differentiable ...

Let $f:\mathbb{R}\to\mathbb{R}$ be an even function. Suppose the left derivative of $f$ at $x=3$ exists and is equal to $-6$. Which of the following is necessarily true?T...

Let $(v_n)$ be a sequence defined by $v_1=2$ and $v_{n+1}=\frac{1}{2}\left(v_n+\frac{8}{v_n}\right)$ for $n\geq1$. If $(v_n)$ converges, then $\lim_{n\to\infty}v_n$ is:$2...

Find the value of $$\int_{-2}^{3}|x^3-3x^2-4x+12|dx$$$\dfrac{125}{4}$ $\dfrac{131}{4}$ $32$ $\dfrac{119}{4}$

The point(s) of inflection of the curve $$y=\frac{2}{5}x^6-\frac{9}{5}x^5-3x^4+22x^3-36x^2+5x+1$$ is/are:$x=-2,1,3$ $x=-2$ and $x=3$ only $x=1$ only None of the above

Evaluate $$I=\int_0^{\frac{\pi}{2}}\frac{\cos x}{\sqrt{1+\sin x}\left(1+\sqrt{1+\sin x}\right)}dx$$$2\ln\left(\frac{1+\sqrt2}{2}\right)$ $\ln\left(\frac{1+\sqrt2}{2}\righ...

At the point $x=1$, the function $f(x)=\begin{cases}(x-1)^2\sin\left(\frac{1}{x-1}\right),&x\neq1\\\\0,&x=1\end{cases}$ is:not continuous continuous but not differ...

Evaluate $$I=\int_0^1 \frac{x^2}{(1-x^2)^{3/2}}dx$$$\frac{\pi}{4}$ $1$ $2$ The integral diverges

Let $f(x)=\left|x-\lfloor x\rfloor-\frac{1}{2}\right|$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. The value of $\int_{0.2}^{2.75} f...

A $20\text{ ft}$ ladder leans against a vertical wall. The base of the ladder is sliding away from the wall at $2\text{ ft/s}$. A point $P$ on the ladder is $6\text{ ft}$...

Let $f:[0,1]\to\mathbb{R}$ be continuous and satisfy $f(x)=x^2+\int_0^1 (x+t)f(t)dt$ for every $x\in[0,1]$. Which of the following is $f(x)$?$x^2-5x-\frac{17}{6}$ $x^2+5x...

Which of the following is the derivative of $f(x)=(x^2+1)^{x\ln x}$ when $x>0$?$(x^2+1)^{x\ln x}\left((\ln x+1)\ln(x^2+1)+\dfrac{2x^2\ln x}{x^2+1}\right)$ $(x^2+1)^{x\ln ...

For a function $f:[-1,1]\to\mathbb{R}$, consider the following statements:There exists $\epsilon>0$ such that for all $x,y\in(-\epsilon,\epsilon)$, if $x<y$, then $f(x)<f...

Define $F,G:\mathbb{R}\to\mathbb{R}$ by $$F(x)=\int_0^x |t(t-2)|dt \qquad\text{ and }\qquad G(x)=\int_0^x F(t)dt$$ Which of the following is true?$G$ is not differentiabl...

Suppose $f:[0,\infty)\to[0,\infty)$ is continuous, strictly increasing, and onto, and let $g=f^{-1}$. For $a,b>0$, define $$I=\int_0^a f(x)dx+\int_0^b g(y)dy-ab$$ Which o...

Evaluate the limit $L=\lim_{x\to\infty}x\left(\sqrt{x^2+6x+10}-\sqrt{x^2+6x+5}\right)$.$0$ $\dfrac{5}{2}$ $5$ $\infty$

The function $f(x)=3x^5-45x^4+230x^3-450x^2+6x+11$ is concave down on which of the following intervals?$(1,3)\cup(5,\infty)$ $(-\infty,1)\cup(3,5)$ $(-\infty,3)\cup(5,\in...

Let $f,g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$, with $f(a)=f(b)=0$. Which of the following must be true?Hint: Apply Rolle’s theorem t...

What are all values of $x$ for which the function $f(x)=\dfrac{x^5}{5}-\dfrac{x^4}{4}-\dfrac{13x^3}{3}+\dfrac{25x^2}{2}-12x+6$ is increasing?$(-\infty,-4)\cup(1,3)$ $(-4,...

For the function $f(x)=x^4-8x^3+18x^2+ax+7$, which statement about having a local maximum at $x=1$ is true?It happens for exactly one value of $a$. It happens for more th...

Let $f,g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Suppose $f'(x)-g'(x)=2x$ for all $x\in(a,b)$. Which of the following must be true?$f(...

A rectangle is inscribed under the curve $y=12-x^2$ and above the $x$-axis, with its base on the $x$-axis and its top corners on the curve. What is the maximum possible a...