Linear regression

Questions:

1. Evaluate the limit

lim h→0

1 4+h

− 1 4

h

2. Consider the function

f(x) =

  x2 − 1 x − 1

if x < 1

c + ln x if x ≥ 1

(a). Evaluate lim x→1−

f(x)

(b). Evaluate lim x→1+

f(x)

(c). Find a value for the parameter c, so that f is continuous at x = 1.

3. Given

f(x) = 4ex + 2e−x

ex − e−x Find all vertical and horizontal asymptotes.

4. Find y′ if xy = yx.

University of Delaware Department of Mathematical Sciences 2021 Fall MATH 241 University of Delaware Department of Mathematical Sciences 2021 Fall MATH 241

5. The population of yeast cells is modeled by the function

n = f(t) = a

1 + be−0.7t

where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of a and b.

6. If a snowball melts so that its surface area decreases at a rate of 1 cm2/min, find the rate at which the diameter decreases when the diameter is 10 cm. (The surface area of a snowball is A = 4πr2).
7. Consider the function f(x) =

x

x2 + 1

(a). Find the intervals on which f is increasing and decreasing.

(b). Find the intervals on which f is concave up and concave down.

(c). Find the local maximum and local minimum values of f.

(d). Find the inflection points of f.

8. Use a linear approximation to estimate cos 29◦
9. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into circle. How should the wire be cut so that the total area enclosed is a maximum.
10. In a murder investigation, the temperature of the corpse was 32.5◦C at 1:30PM and 30.3◦C an hour later. Normal body temperature is 37.0◦C and the temperature of the surroundings was 20.0◦C. When did the murder take place?

Sample Solution