Unit 2: Differential calculus and its applications - Subjective Questions

Standard derivatives:

• Algebraic function:
• Exponential functions:
• Logarithmic functions:
• Trigonometric functions:
• Inverse trigonometric functions:

Let and be differentiable functions.

• Sum rule:
• Difference rule:
• Product rule:
• Quotient rule:
• Chain rule: If and , then

For example, if , the chain rule gives

The function is a product of , , and .

Using the product rule for three functions,

Therefore,

Taking as a common factor,

Equivalently,

For a curve defined parametrically,

Here,

Hence,

The second derivative is

Since

we obtain

Thus,

Implicit differentiation is used when is not explicitly expressed as a function of . Both sides of the relation are differentiated with respect to , treating as a function of .

Given

differentiate with respect to :

Using the product rule,

Therefore,

Collecting terms containing ,

Hence,

Logarithmic differentiation is useful when the variable appears in both the base and the exponent.

Given

take the natural logarithm on both sides:

Differentiate implicitly with respect to :

Multiplying by and substituting gives

The condition ensures that is defined in the real number system.

Take the natural logarithm of both sides:

Differentiate with respect to :

Thus,

Substituting the original expression for ,

Logarithmic differentiation converts products, quotients, and powers into sums and constant multiples, simplifying the calculation.

For , successive differentiation gives

Therefore, the pattern is

For , successive derivatives are

The derivatives repeat after every four differentiations. A compact expression is

Rolle's theorem: If a function satisfies the following conditions:

• is continuous on ,
• is differentiable on ,
• ,

then there exists at least one such that .

For on :

• Since is a polynomial, it is continuous on and differentiable on .
• The endpoint values are
• Thus, .

Now,

Setting ,

Since , Rolle's theorem is verified, with

Lagrange's mean value theorem: If is continuous on and differentiable on , then there exists at least one such that

For on , the function is continuous and differentiable because it is a polynomial.

The average rate of change is

Also,

By the theorem,

so

Since , the theorem is verified, and

Cauchy's mean value theorem: Let and be continuous on and differentiable on . If for every , then there exists at least one such that

Equivalently,

To obtain Lagrange's mean value theorem, choose

Then and . Cauchy's theorem becomes

which is exactly Lagrange's mean value theorem.

Thus, Lagrange's theorem is a special case of Cauchy's theorem.

Suppose possesses derivatives up to order in an interval containing and . Taylor's theorem states that

The Lagrange form of the remainder is

where lies between and .

• is the point about which the function is expanded.
• The finite sum is the Taylor polynomial of degree .
• is the error made by replacing with that polynomial.
• The value determines the coefficient of .

If as , the corresponding infinite Taylor series represents in the interval under consideration.

Maclaurin's theorem is Taylor's theorem with :

For , every derivative is . Hence,

Therefore,

Using terms up to and substituting ,

Thus,

Hence,

For , the derivatives repeat as

At , their values are

Therefore,

For , the derivatives repeat as

At , their values are

Therefore,

The sine series contains only odd powers, whereas the cosine series contains only even powers.

An indeterminate form is an expression obtained during limit evaluation whose appearance alone is insufficient to determine the value of the limit.

The common indeterminate forms are

For example, the form may produce different limits:

while

Both expressions have the form under direct substitution, but their limits differ. Therefore, an indeterminate form is not the final value of the limit. The expression must first be transformed using factorization, rationalization, standard limits, series expansions, logarithms, or L'Hospital's rule.

L'Hospital's rule: If and are differentiable near a point and the quotient has the indeterminate form or , then, under the required regularity conditions,

provided the limit on the right exists.

Consider

Direct substitution gives . Applying L'Hospital's rule once,

This is again , so apply the rule a second time:

Thus,

The expression has the indeterminate form . Let

Taking natural logarithms,

As , this quotient has the form . Applying L'Hospital's rule,

Simplifying,

Therefore,

Hence,

Taking logarithms converts a power-type indeterminate form into a product or quotient form to which L'Hospital's rule can be applied.

A point is a critical point if or if does not exist while is defined.

First derivative test:

• If changes from positive to negative as passes through , then is a local maximum.
• If changes from negative to positive, then is a local minimum.
• If does not change sign, then is neither a local maximum nor a local minimum.

Second derivative test: If , then

• implies that is a local minimum.
• implies that is a local maximum.
• makes the test inconclusive; the first derivative test or higher derivatives must then be used.

The first derivative test examines monotonicity, while the second derivative test examines the concavity of the graph at a stationary point.

Differentiate the function:

The stationary points satisfy :

The second derivative is

At ,

so gives a local maximum. Its value is

At ,

so gives a local minimum. Its value is

Therefore,

and

Let the side of the square base be metres and the height be metres.

The volume condition is

so

Since the tank is open at the top, its surface area is

Substituting for ,

Differentiate:

For a stationary point,

Thus,

Then

Also,

and

Therefore, the surface area is minimized when

The required tank has a square base and a height of .