Exercises

Here are a few problems designed to exercise your human–machine interface.

1.
Verify the formula ${\frac{{d}}{{dx}}}$$\left(\vphantom{ x^{8}}\right.$x8$\left.\vphantom{ x^{8}}\right)$ = 8x7 by starting with the definition of derivative and choosing submenu items such as Expand and Simplify.BITMAPSETAnswer0.2214in0.205in0ina1

2.
Use Newton's method on the function f (x) = x2 + 1, starting with x0 = 0.5. What conclusions can you draw?BITMAPSETAnswer0.2214in0.205in0ina2

3.
Find the equation of one line that is tangent to the graph of

f (x) = x(x - 1)(x - 3)(x - 6)

at two different points.BITMAPSETAnswer0.2214in0.205in0ina3

4.
For 0 < k < 1, the elliptic integral E = $\int_{{0}}^{{\pi /2}}$$\sqrt{{1-k\sin ^{2}t}}$dt has no elementary solution. Use a series expansion of the integrand to estimate E. BITMAPSETAnswer0.2214in0.205in0ina4

5.
Find all the solutions to xy = yx for unequal positive integers x and y.BITMAPSETAnswer0.2214in0.205in0ina5

6.
Blood flowing through an artery flows fastest at the center of the artery, and slowest near the walls of the artery where friction is a factor. In fact, the velocity is given by the formula v(r) = α(R2 - r2), where α is a constant, R is the radius of the artery, and r is the distance from the center.

Set up an integral that gives the total blood flow through an artery. Show that, if an artery is constricted to one-half of its original radius, the blood flow (assuming constant blood pressure) is reduced to ${\frac{{1}}{{16}}}$ of its original flow.BITMAPSETAnswer0.2214in0.205in0ina6

7.
The mass of an object traveling at a velocity v with rest mass m0 is given by

m = m0$\displaystyle \left(\vphantom{ 1-\frac{v^{2}}{c^{2}}}\right.$1 - $\displaystyle {\frac{{v^{2}}}{{c^{2}}}}$$\displaystyle \left.\vphantom{ 1-\frac{v^{2}}{c^{2}}}\right)^{{-1/2}}_{}$

where c is the speed of light. Use a Maclaurin series expansion to show the increase in mass at low velocities.BITMAPSETAnswer0.2214in0.205in0ina7

8.
Evaluate $\int$2xcos bxdx and simplify the answer.BITMAPSETAnswer0.2214in0.205in0ina8

9.
Evaluate $\dint_{{-\pi }}^{{\pi }}$${\dfrac{{1+\sin x}}{{\left(
x-\cos x\right) ^{2}}}}$dx .BITMAPSETAnswer0.2214in0.205in0ina9

10.
Evaluate $\lim_{{h\rightarrow 0^{+}}}^{}$$\int_{{0}}^{{\infty
}}$sin(x1+h) dx .BITMAPSETAnswer0.2214in0.205in0ina10

11.
The Fundamental Theorem of Calculus says that if f is continuous on a closed interval $\left[\vphantom{
a,b}\right.$a, b$\left.\vphantom{
a,b}\right]$, then

a.
If g is defined by g(x) = $\int_{{a}}^{{x}}$f (t)dt for x$\left[\vphantom{
a,b}\right.$a, b$\left.\vphantom{
a,b}\right]$, then g(x) = f (x), and

b.
If F is any antiderivative of f, then. $\int_{{a}}^{{b}}$f (x)dx = F(b) - F(a).

Demonstrate that these two conditions hold for each of the three functions f (x) = x3, f (x) = xex, and f (x) = sin2x cos x.BITMAPSETAnswer0.2214in0.205in0ina11