Recent questions tagged calculus-&-optimization

Let $$f(x)=\frac{(x-3)(x^2+1)}{|x-3|} \text{ for } x\ne 3$$ If $f$ is extended to be continuous at $x=3$, what should be the value of $f(3)$?$10$ $-10$ $0$ No such value ...

The function $f(x)$ is defined by $$f(x)=\begin{cases}\dfrac{x^2+ax+b}{\sqrt{x+3}},&x\le 1\\\\ \ln(2x-1)+\dfrac{5x}{x+1},&x>1\end{cases}$$ where $a$ and $b$ are constants...

Consider the piecewise function $$f(x)=\begin{cases}\dfrac{e^{Cx}-1+4x}{\sin(2x)},&x<0\\\\ \dfrac{(C^2-1)x^2+7x+3}{2x^2+\sqrt{x^4+1}},&x\ge 0\end{cases}$$where $C$ is a c...

Suppose $p(x)$ is twice differentiable for all real values of $x$. Some values of $p'(x)$ and $p''(x)$ are given in the table below:$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}...

A portion of the graph of an even function $f(x)$ is shown only on the interval $[0,6]$, although $f(x)$ is defined for all real numbers. The graph has a vertical asympto...

Evaluate the following limit and find the value of $a$:$$\lim_{x\to 0}\frac{\sqrt{1+4x+7x^2}-\sqrt[3]{1+6x+9x^2}+x-\ln(1+x)}{x^2}=\frac{a}{6}$$What is the value of $a$?

Let $f$ be a continuous function defined for all real numbers. Suppose :$$\int_0^3 f(x)dx=4, \qquad\int_0^6 f(x)dx=9, \qquad\int_6^{10} f(x)dx=-2, \qquad \text{and} \qqua...

Evaluate the definite integral:$$I=\int_0^\pi x\sin x\ln(1+\cos^2 x),dx$$Which of the following is the correct value?$\frac{\pi^2}{2}+\pi\ln 2-2\pi$ $\frac{\pi^2}{4}+\pi\...

Determine the following limit:$$L=\lim_{x\to\infty}x^2\left(\sqrt{9x^2+6x+5}-3x-1-\frac{2}{3x}\right)$$Which of the following is the correct value of the limit?$0$ $\frac...

Suppose that a function $f$ is defined by $$f(x)=\begin{cases}\frac{x^2-2x-8}{x-4}+3, & x\lt 4 \\\\ 25, & x=4 \\\\ \frac{\sqrt{x+5}-3}{x-4}, & x\gt 4\end{cases}$$Let $M=\...

Let $f(x)$ be continuous on $[1,3]$ and differentiable on $(1,3)$. If $f(1)=5$ and $2x+1\le f'(x)\le 4x+3$ for all $x\in(1,3)$, then the smallest and largest possible val...

The value of the given integral is : $$I=\int_0^{\pi/2}\left(\frac{4\sqrt{\sin x}+7\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}+\sin x\cos x\right)dx$$$\frac{11\pi}{4}+\fr...

Evaluate the limit $\lim_{x\to-\infty}(\sqrt{9x^2+6x+5}+3x)$.$-1$ $0$ $1$ Limit does not exist

Consider a differentiable function $f$ whose derivative is given by $f'(x)=e^{-x}(x+2)^2(x-1)(x-3)^3$. Which of the following statements are true?There is a local maximum...

Let $f$ be not defined at $x=2$. For $x\ne 2$, suppose $f(x)=\frac{x^2+ax+b}{x-2}$. Which condition on $a$ and $b$ makes $\lim_{x\to 2}f(x)$ exist as a finite real number...

Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable at $x=x^*$. If $f'(x^*)=0$, which of the following statements must be true?$f$ is continuous at $x=x^*$ The tangent line...

Find the minimum value of $f(x,y)=xy+x+y$ subject to $x^2+y^2=1$.$-1$ $-\sqrt{2}$ $-1/2$ $-2$

Choose the largest interval of $x$ on which the function $f(x)=x^2\ln x$, where $x>0$, is convex.$(0,e^{-3/2})$ $[e^{-3/2},\infty)$ $(0,\infty)$ $[e^{-1},\infty)$

A company splits $12$ research hours between two tasks. If $x$ hours are spent on task A, then $12-x$ hours are spent on task B. The total difficulty score is $D(x)=x^2+(...

A research team has $60$ hours to divide between technical validation and market validation. If $x$ hours are spent on technical validation, then $60-x$ hours are spent o...

A company models its total production cost by $C(x)=x^2+\frac{16}{x}$, where $x>0$ is the number of units produced in hundreds. Find the minimum possible value of $C(x)$.

Determine the relative minimum value of the function:\begin{aligned}&f(x)=x^3-6 x^2+5\end{aligned}

Determine where the function below is NOT CONTINUOUS:$$h(t)=\frac{4 t+10}{t^2-2 t-15}$$$\mathrm{t}=3$ AND $\mathrm{t}=-5$ $\mathrm{t}=5$ AND $\mathrm{t}=-3$ $\mathrm{t}=-...

Estimate the value of the following limit:$$\lim _{x \rightarrow 2} g(x)$$where the function $g(x)$ is defined as:$$g(x)= \begin{cases}\frac{x^2+4 x-12}{x^2-2 x} & \text ...

Given the function $f(x, y)=\frac{x^2}{y^3}$, determine if $f(x, y)$ is INCREASING or DECREASING at the point $(2,5)$, if we allow $x$ to vary and hold $y$ fixed. INCREAS...

Given the function $f(x, y)=\cos (2 x)-x^2 e^{5 y}+3 y^2$, find all the second-order partial derivatives $\left(f_{x x}, f_{y y}\right.$, and $\left.f_{x y}\right)$ : $f_...

Find the minimum value of the function $f(x, y)=4 x^2+10 y^2$ on the disk $x^2+y^2 \leq 4$.$4$ $16$ $0$ $-4$

Find the maximum value of the function $f(x, y)=5 x-3 y$ subject to the constraint $x^2+ y^2=136$.$136$ $-68$ $68$ $34$

Suppose that the height of a hill above sea level is given by $z=1000-0.01 x^2-0.02 y^2$. If you are at the point $(60,100)$ in what direction is the elevation changing f...

$\text { Find the Taylor Series for } f(x)=x^4 \mathrm{e}^{-3 x^2} \text { about } x=0 \text {. }$ $\sum_{n=0}^{\infty} \frac{(-1)^n 3^n x^{2 n+4}}{(n+1)!}$ $\sum_{n=0}^{...