Question

If (1,10) lies on the graph of \(y=f(x)\) and \(f^{\prime}(1)=5,\) what can be said about \(y=f^{-1}(x)\) ?

Short answer

The derivative of the inverse function at 10 is \(\frac{1}{5}\).

Step-by-step solution

Understand the Function and Its Inverse

The problem states that the point \((1,10)\) lies on the graph of \(y=f(x)\). This means that when \(x=1\), \(f(1)=10\). The inverse function \(f^{-1}(x)\) reverses this pair; so, \(f^{-1}(10)=1\).

Derive the Relationship Between Derivatives

The derivative \(f'(1)=5\) gives the slope of the tangent to the graph of \(y=f(x)\) at \(x=1\). For inverse functions, the relationship \(\left(f^{-1}\right)'(y)=\frac{1}{f'(x)}\) holds, where \(y=f(x)\).

Apply the Derivative of the Inverse Function

Since \(f(1)=10\), we can use the inverse function derivative formula at \(y=10\): \(\left(f^{-1}\right)'(10)=\frac{1}{f'(1)}=\frac{1}{5}\). Thus, the slope of the tangent to \(y=f^{-1}(x)\) at \(x=10\) is \(\frac{1}{5}\).

Key concepts

Function Derivative

Understanding the concept of a function derivative is crucial in studying calculus. A derivative measures how a function's output value changes as its input changes. Think of it as the "rate of change" or how fast something is happening. It tells us the slope of the tangent line to the function's graph at any given point.
When we calculate the derivative of a function, symbolized as \(f'(x)\), we are finding the function that gives us these slopes at every point. For instance, in our exercise, \(f'(1) = 5\) tells us that at \(x = 1\), the function \(f(x)\) changes at a rate of 5. This means that at \(x = 1\), the graph of \(f(x)\) will rise 5 units vertically for every 1 unit it moves horizontally.
Understanding derivatives helps analyze dynamics like speed, growth, and spatial changes.

Tangent Slope

The slope of a tangent line is an essential geometric interpretation of a derivative. A tangent line is a straight line that touches a curve at a single point without crossing it at that location. The slope of this line indicates how steep the curve is at that point.
Imagine standing on a hill (the curve) and observing how steep it is. The tangent slope would tell you how much effort you need to move upward. For our function \(f(x)\), at \(x = 1\), the tangent slope is 5, meaning the hill rises steeply at this precise spot.

• If the slope is positive, as it is here, the function is increasing at that point.
• A negative slope would mean the function is decreasing.
• A slope of zero would mean the function is flat at that point.

Knowing the tangent slope is vital for understanding the behavior of functions, especially in dynamics and optimization.

Inverse Function Derivative

The derivative of an inverse function brings nuances of calculus into play. When you have a function \(f(x)\) and its inverse \(f^{-1}(x)\), there's a unique relationship between their slopes. If you're given \(f'(x)\), the derivative of its inverse at \(y\) (where \(y=f(x)\)), is \((f^{-1})'(y) = \frac{1}{f'(x)}\).
This relationship is crucial. It indicates that the steeper \(f(x)\)'s slope at a point, the flatter the inverse function will be at the corresponding point. In the exercise, \((f^{-1})'(10) = \frac{1}{5}\), suggesting that where \(f(x)\) has a steep incline, \(f^{-1}(x)\) reacts less sharply with a lower slope of \(\frac{1}{5}\).
Understanding this inverse relationship is vital in multiple fields, from physics, where inverse functions model reciprocal processes, to economics, where they depict inverse relationships like supply and demand.

Graph of a Function

The graph of a function is a visual representation of how a function behaves across different inputs. It's a snapshot of where each input is transformed into an output, shown across a plane. In our context, the point (1,10) on the graph of \(y=f(x)\) informs us that when \(x=1\), the function outputs 10.
Inverting this graphically involves mapping outputs back to inputs. For \(f^{-1}(x)\), the roles of x and y are switched. The point \((10, 1)\) on \(f^{-1}(x)\)'s graph tells us that \(f^{-1}(10)=1\), showing the reverse relationship.
Graphically, the function and its inverse can be mirrored across the line \(y=x\). This visual flip helps understanding the bidirectional relationship offered by inverses. Knowing how to graph these functions aids in interpreting data and seeing real-world trends visually.