0. Functions

Functions

In Exercises 1–6, find the domain and range of each function.

f(x) = 1 + x²

61–62. Points of intersection and area

a. Sketch the graphs of the functions f and g and find the x-coordinate of the points at which they intersect.

f(x) = sech x, g(x) = tanh x; the region bounded by the graphs of f, g, and the y-axis

Daylight function for 40 °N Verify that the function $D(t)=2.8\(\sin\)(\(\frac{2\pi}{365}\)(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.

$D\left(81\right)=12$ and $D\left(264\right)\approx 12$  (corresponding to the equinoxes).

Functions and Graphs

Find the natural domain and graph the functions in Exercises 15–20.

f(x) = 1 − 2x − x²

Find the area of the region enclosed by one loop of the curve $r=\sin \left(4\theta \right)$.

Which of the following statements is true about the function $g$ defined on the closed interval $[-4,8]$?

Evaluate the triple integral of $1$ over the part of the ball defined by $x^2+y^2+z^2\le 8$ that lies in the first octant.

Evaluate the definite integral: ${\int }_0^1\frac{2}{4}{t}^3-\frac{3}{4}{t}^2+\frac{2}{5}tdt$ from $t=0$ to $t=1$.

Evaluate the integral: ${\int }_0^{\frac{\pi }{2}}\frac{\cos \left(t\right)}{1+{\sin }^t^2}dt$

Evaluate the integral: ${\int }_{x=1}^{x=2}2x(x^2+4x+20) dx$.

Evaluate the iterated integral by converting to polar coordinates: $${\int }_0^2{\int }_{2-x^2}^{4-x^2}{x}^2+y^2 dy dx$$

Find the exact length of the curve given by $x=\frac{y^4}{8}+\frac{1}{4y^2}$, for $1\le y\le 3$.

Is the constant function $y\left(t\right)=-4$ a solution to the differential equation $y^{\prime }=t^2(y+4)$?

Which of the following is a power series representation for the function $f\left(x\right)=x^6\arctan \left(x^3\right)$ centered at $x=0$?

Which of the following is a power series representation for the function $f\left(x\right)=x^4\arctan \left(x^3\right)$ centered at $x=0$?