Math Calculus questions.

Math 251 Written Homework 2 - Spring 2018 Due Thurs. April 19th Page 1 of 2

Name: Student ID#:

Instructions: Put your solutions on a separate piece(s) of paper (8.5x11 or A4). Use this page as a cover sheet. Staple all of the pages together. Late homework is not accepted. Turn in your homework during recitation.

(1) Evaluate the following limit.

lim t→3

[( 4t−

2

t− 3

)( 6 + t− t2

)]

(2) Give a formula for a function f that satisfies the following conditions. (The work you can show for this problem is: Show that your function satisfies all of the conditions.)

• lim x→4+

f(x) = −∞

• lim x→4−

f(x) = ∞

• f(0) = 0

(3) The amount of an antibiotic (in mg) in the blood t hours after an intravenous line is opened is given by m(t) = 100(e−0.03t −e−0.1t).

(a) Use the intermediate value theorem to show that the amount of the drug is 30 mg at some time in the interval [0, 10] and again sometime in the interval [10, 50].

(b) Is the amount of the drug in the blood ever 40 mg? Why or why not?

Math 251 Written Homework 2 - Spring 2018 Due Thurs. April 19th Page 2 of 2

(4) Let g(x) =

  x2 + x x < 1

a x = 1

3x + 5 x > 1.

(a) Determine the value of a for which g is continuous from the left at 1. (b) Determine the value of a for which g is continuous from the right at 1. (c) Is there a value of a for which g is continuous at 1? Explain.

(5) Find the derivative f ′ of the function f(x) = x − x2 using the definition of the derivative. Then find the equation of the tangent line at x = 2.