Unit 4: Calculus - Subjective Questions

Definition: The statement means that the values of can be made arbitrarily close to by taking sufficiently close to, but not equal to, .

- definition: For every , there exists a such that

Here, represents the required closeness of to , while represents the corresponding closeness of to .

Important point: The value of need not be equal to for the limit to exist.

The standard trigonometric limit is

Explanation: For , the geometrical inequality

holds. Dividing by gives

Taking reciprocals,

As , both and approach . Therefore, by the squeeze theorem,

The same result holds for approaching zero from the left.

A function is continuous at if and only if all the following conditions are satisfied:

• is defined.
• The limit exists.
• The limit is equal to the value of the function:
  

Equivalently, continuity requires

If any one of these conditions fails, the function is discontinuous at . A function is continuous on an interval if it is continuous at every point of that interval.

For continuity at , the left-hand limit, right-hand limit, and function value must be equal.

The left-hand limit is

The right-hand limit and function value are

Equating these values,

Therefore,

Answer: The function is continuous at when

For this value, both one-sided limits equal , and .

Differentiability: A function is differentiable at if the derivative

exists as a finite number.

Relationship:

• If a function is differentiable at , then it is necessarily continuous at .
• A function may be continuous at without being differentiable there.

To prove the first statement, write

As , the first factor tends to and the second tends to . Hence

which implies .

For example, is continuous at but not differentiable there because its left and right derivatives are different.

Let

Taking natural logarithms on both sides,

Differentiating with respect to ,

Therefore,

Since ,

The condition ensures that is defined.

Rolle's theorem: If a function is continuous on , differentiable on , and satisfies , then there exists at least one such that

For :

• It is a polynomial, so it is continuous on and differentiable on .
• .
• .

Thus, . Its derivative is

Setting the derivative equal to zero,

Since , Rolle's theorem is verified.

Mean Value Theorem: If is continuous on and differentiable on , then there exists at least one point such that

The expression on the right is the slope of the secant line joining and . The derivative is the slope of the tangent at .

Geometrical interpretation: At some interior point , the tangent to the curve is parallel to the secant line joining the endpoints.

Rolle's theorem is a special case of the Mean Value Theorem when .

Assume first that . The function is continuous on and differentiable on . By the Mean Value Theorem, there exists such that

Taking absolute values,

Since ,

If , interchanging and gives the same result. Hence the inequality holds for all real and .

The first derivative test uses the sign of around a critical point .

• If changes from positive to negative at , then changes from increasing to decreasing. Therefore, is a local maximum.
• If changes from negative to positive at , then changes from decreasing to increasing. Therefore, is a local minimum.
• If the sign of does not change at , then the test does not give a local maximum or minimum.

Critical points occur where

or where does not exist, provided is defined.

The test is applied by constructing a sign chart for on intervals around each critical point.

Differentiate the function:

The critical points are

The sign of is:

• Positive for .
• Negative for .
• Positive for .

Thus, changes from increasing to decreasing at , so there is a local maximum there. Its value is

At , the function changes from decreasing to increasing, so there is a local minimum. Its value is

Answer:

Suppose and exists.

• If , the graph is concave upward near , and is a local minimum.
• If , the graph is concave downward near , and is a local maximum.
• If , the test is inconclusive, and the first derivative test or another method must be used.

For example, if and , then the function has a local minimum at .

Let the rectangle have length and width . Since its diagonal is the diameter of the circle,

The area is

Solving for ,

Thus,

A simpler argument uses the inequality

Since ,

Therefore,

Equality holds when , so the rectangle is a square.

Answer: The rectangle of maximum area is a square with side length

and maximum area

An indefinite integral of is the family of all functions whose derivative is . It is written as

where

The symbol is called the constant of integration. It is included because the derivative of every constant is zero. For example,

and

Therefore,

The indefinite integral represents a family of curves differing only by vertical translations.

Let

Then

so

Substituting,

Therefore,

Replacing by gives

Start with the product rule:

Multiplying by gives

Rearranging,

Integrating both sides,

Hence, the integration by parts formula is

It is useful when the integrand is a product of two functions, such as a polynomial multiplied by an exponential, logarithmic, or trigonometric function. The choice of is commonly guided by the priority order logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential functions.

Choose

Then

Using integration by parts,

Therefore,

Thus,

The result is valid for , where is defined.

The Fundamental Theorem of Calculus has two main parts.

Part I: If is continuous on and

then

Thus, differentiation reverses the accumulation represented by integration.

Part II: If is any antiderivative of , so that , then

Therefore, a definite integral can be evaluated by finding an antiderivative and subtracting its values at the upper and lower limits.

An antiderivative of is . Therefore,

Evaluating the limits,

Hence,

Geometrically, this value represents the area under the curve

above the -axis and between and .

Continuity and differentiability are related but distinct properties.

Continuity: A function is continuous at if

It means that the graph has no break, jump, or hole at that point.

Differentiability: A function is differentiable at if the finite derivative

exists. It means that the function has a well-defined tangent at that point.

Comparison:

• Differentiability at a point implies continuity at that point.
• Continuity does not imply differentiability.
• The function is continuous at , but its left derivative is and its right derivative is , so it is not differentiable at .
• The polynomial is both continuous and differentiable everywhere.