School of Mathematics and Statistics
Carleton University
Math. 69.107
ASSIGNMENT 1
SOLUTIONS
Solution From the Binomial Theorem we know that . Next, we have or, since f(1) = 3-2+1 = 2, we have for ,
It follows that
Remark: As a check note that by the Power Rule.
At x=0 we have an indeterminate form, 0/0, so we must rewrite the expression in order to evaluate the two-sided derivative as an ordinary limit. Note that
for x>0. Using this we see that the right-derivative of f at x=0 is given by
The same argument can be used to show that
if x < 0 since , in this case. So,
Since the left and right-derivatives are not equal at x=0, it follows that DOES NOT EXIST.
Find the equation of the tangent line to the graph of y = f(x) at the point .
Solution y is given implicitly. So, the slope of the required tangent line is given by where is given by
Solving for , we get
Thus, . It follows that the tangent line looks like y-0 = 0(x-1) = 0. So y=0 (or the x-axis) is the equation of the tangent line.
In order to evaluate at the point we know that
At x=1, y=0 we get that .
Solution a) ,
b)
Solution a) , since as . On the other hand, as . So their product must approach .
Alternately,
b) We use the identity with . Then,
But as . Furthermore as and so
On the other hand,
It follows that the required limit DOES NOT EXIST.
Let's assume that a planetary body is orbiting the sun (which is assumed to be very close to the origin) in a fixed almost circular orbit given by
Let's say that an asteroid is approaching the sun in an orbit whose equation is given by
Use Newton's method to find the two expected points of crossing of these two orbits, i.e., the two possible collision points, to three significant digits.
Solution Use . Let and . The required roots are given by the zeros of h(x) = f(x) - g(x). Find and then . Then use the iteration
and the answer is 0.995 where there are two values for y given by .
Total: /60