Answer:
a) for k≤0 , h has no critical point
b) for k>0 , h has a critical point
c) for k=0 , has a horizontal asymptote
Step-by-step explanation:
for the function
h(x)=e^(−x)+k
h has a critical point when the first derivative is =0 or is undefined. Since e^(−x) and k*x are continuos functions for all x then the second case is discarded. Then
dh/dx = -e^(−x)+k = 0
k = e^(−x)
x = ln (1/k)
since ln (1/k) should be possitive then k should be >0 . Thus h(x) has a critical point when k>0 and do not have any when k≤0
h has a horizontal asymptote when
lim h(x)=a when x→∞ (or -∞)
then
when x→∞, lim h(x)= lim e^(−x)+k*x = lim e^(−x) + k* lim x = 0 + k*∞ = ∞
on the other hand , when k=0 , lim h(x)= lim e^(−x)= 0 , then h has a horizontal asymptote for k=0
for x→(-∞) , e^(-x) rises exponentially , thus there is no k such that h has an horizontal asymptote when x→(-∞)
The mean is the average of the set. While the mode is the most common number and the median is the middle number of the data set. Hope this helps!
The dependent variable is the number of times you have typed. This is because time is the independent (x) variable as the number of times you've typed is the dependent (y) variable.
1. This probability is 1/2*1/2=1/4, or 0.25.
2. There are three ways to do this: draw two 2s, a three then a two, or a two then a three. So our probability is 3/16=.1875.
3. We already calculated the probability that the sum is less than 6. Using similar methods, we calculate that there are three ways for a sum to be greater than 8, so our probability is 6/16=0.375.
<u>Given</u>:
The sides of the base of the triangle are 8, 15 and 17.
The height of the prism is 15 units.
We need to determine the volume of the right triangular prism.
<u>Area of the base of the triangle:</u>
The area of the base of the triangle can be determined using the Heron's formula.
Substituting a = 8, b = 15 and c = 17. Thus, we have;
Using Heron's formula, we have;
Thus, the area of the base of the right triangular prism is 36 square units.
<u>Volume of the right triangular prism:</u>
The volume of the right triangular prism can be determined using the formula,
where 
is the area of the base of the prism and h is the height of the prism.
Substituting the values, we have;
Thus, the volume of the right triangular prism is 450 cubic units.
