Solution

Solution:
1.(a) This rule can be apply to solve limits where (f(x))/(g(x)) form 0/0 , ?/?, or n/0 where n ? 0. By applying the principle of L’ Hospital’s rule lim?(n?a)??((f(x))/(g(x)))^ ?=lim?(n?a)??(f^i (x))/(g^i (x))? i.e. The numerator and the denominator can be differentiating separately. This process can be reiterated as many times as possible until a definite result is obtained. Note that f^i (x) can be a derivative of f(x)and g^i (x) can also be a derivative of g(x).
Its Application: it can be used to find the limiting values where the derivatives of both numerators and denominators can be found easily.
(b) Relationship between Differentiability and Continuity
Continuity:
The differentiable functions can all be continuous but not all the continuous functions can be differentiable.
For a function to be continuous, the following conditions must be meant:
(1) ?lim?? ???(n?a) f(x) This function must exist.

(2) ?lim?? ???(n?a) f(x)=f(b) This function must exist for all points b.

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Differentiability: If the derivative of the function f at the point b in its domain is given by

f ?(b) = lim

h?0 f(b + h) ? f(b)
h

Therefore, the function is considered differentiable at the point b in its domain if f ? (b) exists.
Differentiability Graph

Continuity Graph

(c)
1. The function is good and smooth and also both differentiable and continuous.
2. There is discontinuity in the function therefore it is not differentiable.
3. The function is not differentiable because of the sharp “point”
4. The function is discontinuous therefore it cannot differentiate.
5. The function is not differentiable because of the pointed corner on the curve
(d)
1. To find both the maximum and minimum parameters of particular function such as cost, profit, loss, amount of the building material used
2. It can also be used in modeling the behavior of moving object
3. The electronic version of the odometer and speedometer in automobile used the concept of derivative to transform the data sent to the electronic motherboard from the tires to miles per Hour (MPH) and distance
4. The radar gun used by police officers uses the concept of derivative to calculate the speed at which the car was going and also report the distance at which the car was from the radar gun
5. The concept are also used by the Government in population censuses
(e)
Vertical Line Test: If there are no two different points in graph that have the same first coordinates, this implies that the vertical lines cross the graph just at once. This is referring to as a vertical line test that is, the graph that passes the vertical line test is a graph of a function.

Its Application: To determine whether a relation is a function or not

Horizontal Line Test: If there are no two different points appearing in a graph that have the same second coordinate, this implies that the horizontal lines cross the graph just at once. This can be refers to as horizontal line test that is one-to-one function.

Its Application: To determine whether if a function is one-to-one.