Part 1:
1.
This notation means that the limit of a function f(x) as x tends to a is equal to L. In this case, L is a number. a is a number. x is the function. No we cannot conclude anything about f (a).
2.
The three conditions associated with non-existence of a limit are
- Divergence up to infintiy
- Oscillations with a frequency increasing with time
- Scintillation
3.
Limx-a f(x) = ∞means that the limit of the function as x tends to a is infinity while Limx-∞ f(x) = b means that as the limit of the function as x tends to infinity is equal to b. Yes, each of the limits exists since the function assumes a certain value as it tends towards a specific value.
4.
An asymptote of a curve is defined as a line whereby the distance between the line and a curve tends to zero at infinity.
For the function provided when x = a then h (a) = 0 and h (x) = 0. Therefore, f(x) = g (x) / 0. This function is undefined since we cannot be able to divide through by zero and therefore will have a vertical asymptote.
y = 0 is an example of a function with a horizontal asymptote.
5.
The function is used in the determination of derivatives from first principles. The function defines the derivative. It gives a function since it is used to determine the first derivative of a function.
6.
A function that is continuous at a point a when the limit to the left is equal to the limit to the right. The function also has a value at point a.
On the other hand, the function is differentiable at a if its gradient can be determined at the point.
An example is the absolute value function at x = 0.
Yes, such a function is also continuous since differentiability indicates that the function is continuous.
7.
An explicit function is of the form y = f(x) implying that is only has x in the function while an implicit function is of the form a = f(x, y) implying that it has both x and y in the function.
Implicit differentiation works when the function a = f(x, y) is differentiated with respect to x while still in its implicit form.
Logarithmic differentiation works as follows:
For f(x) = loga (x) the f’(x) = 1 / (x log (b)).
It is used when there are powers or logs in the function provided.
8.
A local maximum is the highest value that a function attains at a point.
An absolute maximum is the highest value that the function can attain for all points.
9.
An antiderivative is the indefinite integral of a function.
A definite integral is an integral where the upper and lower limits have been defined.
A indefinite integral is an integral where the upper and lower limits have not been defined.
The three concepts are connected since they relate to the integration of functions.
10.
- When a limit does not exist the derivative does not also exist
- When a limit does not exist the derivative does not also exist
- A derivative is a limit
- Integrals are the opposite of derivatives
- Derivatives are the opposite of integrals
Part 2:
1.
Yes, I am hapy with my performance in calculus this semester.
2.
I was able to understand all topics since the lectures received were adequeate. If I was to change anything, I would do more practice outside class in order to improve my calculus skills.
3.
This semester I tried studying using group work and it worked excellently. This is mainly because during group discussions people helped each other easily understand the concepts being introduced in class. For any future calculus student I would recommend additional research outside the classroom.
4.
Everything learnt in class was math related.
