Problem 17 - Calculus

Question

True (with reason) or false (with example): (a) If \(f(t)<0\) then \(d f / d t<0\) (b) The derivative of \((f(t))^{2}\) is \(2 d f / d t\) (c) The derivative of \(2 f(t)\) is \(2 d f / d t\). (d) The derivative is the limit of \(\Delta f\) divided by the limit of \(\Delta t\).

Step-by-Step Solution

Answer

(a) False, (b) False, (c) True, (d) False.

Step 1: Analyzing Statement (a)

The statement is "If \(f(t) < 0\), then \(df/dt < 0\)." This claim implies that for a function with a negative value, its derivative must also be negative. This is false because the sign of \(f(t)\) does not dictate the sign of its derivative. For instance, \(f(t) = -t^2\) is a function where \(f(t) < 0\) for \(t eq 0\), but \(df/dt = -2t\) can be positive, negative, or zero depending on the value of \(t\).

Step 2: Analyzing Statement (b)

The statement is "The derivative of \((f(t))^2\) is \(2 df/dt\)." By applying the chain rule to \((f(t))^2\), we get \(d((f(t))^2)/dt = 2f(t) \cdot (df/dt)\). Therefore, the statement is false by omission of the \(f(t)\) term.

Step 3: Analyzing Statement (c)

The statement is "The derivative of \(2f(t)\) is \(2 df/dt\)." Using the constant multiple rule, if you differentiate \(2f(t)\), you directly apply \(d(2f(t))/dt = 2(df/dt)\). This is true because the constant 2 remains as a factor of the derivative of \(f(t)\).

Step 4: Analyzing Statement (d)

The statement is "The derivative is the limit of \(\Delta f\) divided by the limit of \(\Delta t\)." The derivative is actually defined as the limit of the ratio \(\Delta f/\Delta t\) as \(\Delta t\) approaches zero: \(df/dt = \lim_{\Delta t \to 0} (\Delta f/\Delta t)\). Therefore, it is false to say that it is the division of the individual limits of \(\Delta f\) and \(\Delta t\), should always be considered as a ratio where \(\Delta t\) approaches zero.

Key Concepts

Chain rule

The chain rule is an essential concept in calculus. It allows us to differentiate composite functions, which are functions composed of two or more simpler functions. The chain rule can be thought of as a process to unravel these compositions.
To apply the chain rule, think of one function nested within another, like cases in mathematics where a variable substitution might simplify solving an equation.
If you have a function of the form \(g(f(t))\), its derivative is determined using the chain rule: \(\frac{d}{dt}[g(f(t))] = g'(f(t)) \cdot f'(t)\).

• The outer function is differentiated as normal but keeping the inner function in place.
• The inner function is then differentiated on its own, and its derivative multiplies the derivative of the outer function.

The chain rule is crucial for handling complex derivatives in calculus, simplifying what would otherwise be very complicated problems.

Derivative

The derivative represents the rate of change of a function with respect to its variable. It's the calculus tool used to analyze how a function changes as its input changes.
For a function \(f(t)\), the derivative is denoted as \(df/dt\). It measures how \(f(t)\) changes for an infinitesimal change in \(t\).

• If the derivative is positive, the function is increasing.
• If the derivative is negative, the function is decreasing.

Think of the derivative as a snapshot of the function's behavior at any given point. The concept of a derivative is foundational in calculus, directly leading to advanced topics like those in differential equations and continuous optimization.

Differentiation

Differentiation is the action of calculating a derivative. It's the process of finding how a function changes at any point, visually interpreted as the slope of the tangent line to the function's graph.

• This requires a function to be well-behaved, meaning it must be continuous and smooth where you intend to differentiate it.
• Some primary differentiation techniques include power, product, quotient, and chain rules.

In differentiation, the goal is to find \(\frac{df}{dt}\), giving insight into the instantaneous rate of change rather than just average rate of change. This process is a cornerstone of calculus that helps to study and model real-world phenomena.

Limit

A limit is a fundamental concept in calculus, and it is used to define derivatives. The limit describes the value that a function approaches as the input approaches a specific point.
Understanding limits grants insight into the behavior of functions at points of discontinuity or where the function's behavior sharply changes.
The derivative of a function \(f(t)\) is defined as:\[\frac{df}{dt} = \lim_{\Delta t \to 0} \frac{\Delta f}{\Delta t}\] This describes the derivative as the limiting value of the average rate of change of the function as the interval \(\Delta t\) becomes infinitesimally small. The idea of limits is crucial as it underlies differentiation and helps solve problems regarding continuity and convergence.

Constant multiple rule

The constant multiple rule is a basic but powerful differentiation rule. It allows you to pull constants out of a derivative operation.
If given a function \(f(t)\) scaled by a constant \(c\), the derivative \(\frac{d}{dt}[c \cdot f(t)]\) becomes \(c \cdot \frac{df}{dt}\). This means:

• The constant \(c\) multiplies the derivative just like it multiplies the function.
• This rule simplifies the differentiation process when functions are scaled by constants.

The constant multiple rule is applicable widely in calculus and simplifies calculations by keeping constants separate from the variable-dependent part of functions. Using this rule consistently helps streamline derivative calculations and can help make complex problems appear simpler.