Question

Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. $$ y=-4 x+e^{x} $$

Short answer

The point at which the graph of the function has a horizontal tangent line is at (ln(4), -4*ln(4) + 4).

Step-by-step solution

Find the derivative of the function

In order to find the points where the tangent line of the function is horizontal, first the derivative of the function needs to be found. The derivative of a constant times a variable is a constant, so the derivative of \(-4x\) is \(-4\). The derivative of \(e^{x}\) is \(e^{x}\), hence the derivative of the function is \(-4 + e^{x}\).

Set the derivative equal to zero and solve for x

To find the points where the tangent line is horizontal, set the derivative equal to zero and solve for x. This gives the equation \(-4 + e^{x} = 0\). Solving for x gives \(x = ln(4)\).

Substitute x back into the original function to find y

Substitute \(x = ln(4)\) back into the original equation to find the y-coordinate of the point. This gives \(y = -4*ln(4) + e^{ln(4)} = -4*ln(4) + 4\).

Key concepts

Derivative of a Function

In calculus, the derivative of a function represents the rate at which the function's value changes with respect to changes in its input value. Derivatives are foundational in determining various aspects of a function's behavior, such as the slope of the tangent line to a function's graph at a given point.

Consider the function given in the exercise, \(y = -4x + e^x\). Here, breaking down the function into two separate terms, the first term, \(-4x\), is a linear term. Its derivative is simply \(-4\) because the slope of any linear equation \(ax + b\) is \(a\). The second term, \(e^x\), is an exponential function. The derivative of \(e^x\) is unique because it is equal to itself; that is, the derivative of \(e^x\) is \(e^x\).

Therefore, applying the concept of derivatives to the given function, we combine the derivatives of the individual terms to get the overall derivative of the function, which is \(-4 + e^x\). This derivative tells us how the slope of the tangent line to the graph of the function changes for different values of \(x\).

Solving Exponential Equations

When it comes to solving exponential equations, we often rely on the properties of exponents and logarithms. An exponential equation is one where the variable appears in an exponent; for example, the equation \(a^x = b\) is considered an exponential equation.

To solve such equations, we utilize the fact that logarithms are the inverse operations of exponentiation. In the case of the equation from the step-by-step solution, \(-4 + e^x = 0\), we first isolate the exponential term to obtain \(e^x = 4\). At this point, to determine the value of \(x\), we apply the natural logarithm to both sides of the equation. Since the natural logarithm of \(e^x\) simplifies to \(x\) (because \(\text{ln}(e^x)=x\) due to the inverse relationship), we get \(x = \text{ln}(4)\).

The process of taking logarithms to solve exponential equations is a fundamental technique that allows us to find the values for the variable that satisfy the original equation, which is particularly useful when that variable is part of the exponent.

Natural Logarithm

The natural logarithm, denoted as \(\text{ln}(x)\), is a logarithm with base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm is a special type of logarithm because the base \(e\) is an essential constant in mathematics, particularly in calculus, due to its unique properties in growth processes and continuous compounding.

Properties of natural logarithms play a critical role in both calculus and algebra. In calculus, we use the natural logarithm to integrate functions that appear in the form of \(1/x\). In algebra and exponential equations like the one seen in our exercise, we use the natural logarithm to solve for a variable in the exponent.

When solving the equation \(e^x = 4\), we take the natural logarithm of both sides, which simplifies to \(x = \text{ln}(4)\) since the natural logarithm and the exponential function are inverses of each other. The ability to translate between logarithmic and exponential forms is a powerful tool that effectively simplifies computations and provides solutions to problems involving exponential growth or decay.