Calculus Questions

A piecewise-defined function is defined as:
$$f(x)=\{\begin{array}{ll}-1& \text{if }x\le 0\\ ax+b& \text{if }0<x<1\\ 1& \text{if }x\ge 1\end{array}$$
Where $a$ and $b$ are constants. If the function is continuous everywhere, what is the value of $a$?

What is $\int e^x\left(\sqrt{x}+\frac{1}{2\sqrt{x}}\right)dx$ equal to?

The area enclosed between the curves $y^2=x$ and $y=|x|$ is:

What is the area of one of the loops between the curve $y=c\sin x$ and the x-axis?

Correct evaluation of $\int \frac{x}{(x-2)(x-1)}dx$ (where $p$ is an arbitrary constant) is:

If ${\lim }_{n\to \infty }{\sum }_{k=1}^n\left[{\left(\frac{3k}{n}\right)}^2+2\right]\frac{3}{n}={\int }_0^{6}f(x)dx$, then:

If $y=4x-5$ is tangent to the curve $y^2=p{x}^3+q$ at $(2,3)$, then:

A manufacturing company has its variable cost given by $C(x)=x(80-x^2)$ where $x$ is the number of quantities produced. The price per unit is $p(x)=5x$. The ratio of marginal cost and marginal revenue when 5 units were both produced and in demand is:

If Rolle's theorem holds for the function $f(x)=x^3-a{x}^2+bx-4,x\in [1,2]$ with $f^{\prime }(4/3)=0$, then the ordered pair $(a,b)$ is equal to:

If $x=3\tan t$ and $y=3\sec t$, then the value of $\frac{d^{2}y}{d{x}^2}$ at $t=\frac{\pi }{4}$ is:

Find the area under the curve $y=x+2$, bounded by the x-axis and the y-axis.

The maximum slope of the curve $y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$ occurs at the point:

Let $f$ be a differentiable function such that $f(1)=2$ and $f^{\prime }(x)=f(x)$ for all $x\in \mathbb{R}$. If $h(x)=f(f(x))$, then $h^{\prime }(1)$ is equal to:

Let $f:(0,+\infty )\to \mathbb{R}$ and $F(x)={\int }_0^{x}f(t)dt$. If $F(x^2)=x^2(1+x)$, then $f(4)$ equals: