Problem 2 - Thomas Calculus

Question

solve for \(t.\) \begin{equation} \quad \text { a. }e^{-0.01 t}=1000 \quad \text { b. } e^{k t}=\frac{1}{10} \quad \text { c. } e^{(\ln 2) t}=\frac{1}{2}\end{equation}

Step-by-Step Solution

Answer

a. \(t \approx -690.775\), b. \(t = \frac{\ln(0.1)}{k}\), c. \(t = -1\).

Step 1: Identify the Equation for Part (a)

The equation we are working with for part (a) is \(e^{-0.01 t} = 1000\). Our goal is to solve for \(t\).

Step 2: Take the Natural Logarithm on Both Sides for Part (a)

Apply the natural logarithm to both sides of the equation: \(-0.01t = \ln(1000)\). The natural logarithm \(\ln\) and the exponential function \(e\) cancel each other out.

Step 3: Solve for t in Part (a)

Solve for \(t\) by dividing both sides by \(-0.01\): \(t = -\frac{\ln(1000)}{0.01}\). Calculate this value to find \(t\).

Step 4: Identify the Equation for Part (b)

The equation for part (b) is \(e^{kt} = \frac{1}{10}\). Our goal is to solve for \(t\).

Step 5: Take the Natural Logarithm on Both Sides for Part (b)

Apply the natural logarithm to both sides: \(kt = \ln\left(\frac{1}{10}\right)\), effectively undoing the exponentiation by \(e\).

Step 6: Solve for t in Part (b)

Divide both sides by \(k\) to isolate \(t\): \(t = \frac{\ln\left(\frac{1}{10}\right)}{k}\). Calculate the value of \(t\).

Step 7: Identify the Equation for Part (c)

The equation for part (c) is \(e^{(\ln 2) t} = \frac{1}{2}\). Our goal is to solve for \(t\).

Step 8: Take the Natural Logarithm on Both Sides for Part (c)

Apply the natural logarithm to both sides: \((\ln 2)t = \ln\left(\frac{1}{2}\right)\). The \(e\) and \(\ln\) operations cancel.

Step 9: Solve for t in Part (c)

Divide both sides by \(\ln 2\) to solve for \(t\): \(t = \frac{\ln\left(\frac{1}{2}\right)}{\ln 2}\). Calculate this value.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is an essential concept in mathematics, particularly when dealing with exponential growth and decay. It is the inverse function of the exponential function, meaning it helps to undo exponential effects.

When you come across an equation that features an exponential term such as \(e^x\), applying the natural logarithm to both sides is a common technique to solve for the variable in the exponent. This is because the natural logarithm and the exponential function \(e\) are inverse operations. Applying \(\ln\) to \(e^x\) results in a simplified form, \(x\), because they cancel each other out.

Useful properties of the natural logarithm include:

• \(\ln(e) = 1\)
• \(\ln(1) = 0\)
• \(\ln(a \cdot b) = \ln(a) + \ln(b)\)
• \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)

These properties make the natural logarithm a powerful tool for simplifying complex equations and solving for variables in the exponent.

Exponential Function

The exponential function, represented as \(e^x\), is a fundamental mathematical concept with applications in various fields, from biology to finance. Defined as a constant base \(e\) (approximately 2.71828) raised to the power of \(x\), the exponential function describes growth or decay processes.

The function \(e^x\) exhibits some key characteristics:

• It always produces positive values, no matter what the exponent \(x\) is.
• For \(x > 0\), \(e^x\) describes exponential growth, meaning as \(x\) increases, \(e^x\) increases rapidly.
• For \(x < 0\), \(e^x\) represents exponential decay, where the value approaches zero as \(x\) becomes more negative.

The exponential function is unique because it is its own derivative and its own integral, making it indispensable in calculus and differential equations.

In equations like those provided in the exercise, exponential terms can be managed by applying the natural logarithm to both sides, allowing for the isolation and solving of the exponent.

Solving Equations

When solving exponential equations, the goal is to find the value of the unknown variable that makes the equation true. The process typically involves isolating the variable in the exponent, and often requires the use of the natural logarithm.

Here's a general approach to solving equations involving an exponential function:

• Identify the equation and make sure the exponential term is isolated.
• Take the natural logarithm on both sides of the equation. This step uses the property that \(\ln(e^x) = x\).
• Simplify the equation to find the expression involving the unknown variable.
• Solve for the variable by performing algebraic operations, such as division or multiplication, as required.

In practice, follow these steps clearly. In the given exercise, parts (a), (b), and (c) all use this method to solve for \(t\). Each equation requires careful application of the natural logarithm to unwrap the exponent and solve for the desired variable. Simplifying and calculating these expressions precisely will lead to the correct solution for the problem at hand.