next up previous
Next: About this document

MATH 1300.03F D Exam 1 Friday, October 8, 1999

NAME:

Student Number:


tex2html_wrap_inline191.(5 Marks) Let tex2html_wrap_inline193, tex2html_wrap_inline195.

(i) Show that f is one to one on tex2html_wrap_inline201.

(ii) Find the domain of the inverse function of f.

(iii) Find a formula for the inverse function.

(i) Proof. Assume that f(x)=f(x'), tex2html_wrap_inline213. Then tex2html_wrap_inline215. This implies tex2html_wrap_inline217 and x=x'. Hence, f is one to one on tex2html_wrap_inline201.

(ii) Let tex2html_wrap_inline227. Then tex2html_wrap_inline229. Hence, the domain of the inverse function is tex2html_wrap_inline201.

(iii) tex2html_wrap_inline235, tex2html_wrap_inline195.

tex2html_wrap_inline239. (4 Marks) Evaluate the following limits.

(a) tex2html_wrap_inline243.

(b) tex2html_wrap_inline247

(c) tex2html_wrap_inline251.

(d) tex2html_wrap_inline255.

tex2html_wrap_inline257. (6 Marks) Determine whether each of the following functions is continuous at x=0.

(1) tex2html_wrap_inline263

Soln: We rewrite f as follows:
displaymath267

Then, we have
displaymath269
.

This implies tex2html_wrap_inline271. Hence, f is continuous at x=0.

(2) tex2html_wrap_inline279

Soln: We rewrite f as follows:
displaymath283

Hence, we have
displaymath285

This implies tex2html_wrap_inline287 does not exist and f is not continuous at x=0.




Kunquan Lan
Mon Jan 24 11:36:45 EST 2000