Problem 60 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the function \(f\) given that the slope of the tangent line to the graph of \(f\) at any point \((x, f(x))\) is \(f^{\prime}(x)\) and that the graph of \(f\) passes through the given point. $$f^{\prime}(t)=t^{2}-2 t+3 ;(1,2)$$

Step-by-Step Solution

Answer

The function \(f(t)\) is given by its derivative \(f'(t) = t^2 - 2t + 3\). To find the function, we integrate the derivative and use the given point \((1, 2)\) to find the constant of integration. The final function is: $$f(t) = \frac{1}{3}t^3 - t^2 + 3t + \frac{2}{3}$$

Step 1: Integrate the derivative

We are given the derivative of the function, \(f'(t)=t^2-2t+3\). To find the function \(f(t)\), integrate the derivative with respect to \(t\): $$f(t) = \int (t^2 - 2t + 3) dt$$

Step 2: Evaluate the integral

Finding the antiderivative by applying the power rule of integration: $$f(t) = \frac{1}{3}t^3 - t^2 + 3t + C$$ where C is the constant of integration.

Step 3: Use the given point to find the constant

We are given that \(f(1) = 2\). Plug the point (1, 2) into the equation and solve for the constant of integration, C: $$2 = \frac{1}{3}(1)^3 - (1)^2 + 3(1) + C$$

Step 4: Solve for the constant C

Simplify and solve for C: $$2 = \frac{1}{3} - 1 + 3 + C$$ $$C = 1 - \frac{1}{3}$$ $$C = \frac{2}{3}$$

Step 5: Write the final function

Now that we have found the constant C, we can write the function \(f(t)\) as: $$f(t) = \frac{1}{3}t^3 - t^2 + 3t + \frac{2}{3}$$

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with rates of change. It helps us understand how functions change at specific points and is key for finding slopes and tangents. If you think of a function as a curve on a graph, differentiation allows you to calculate the slope of this curve at any given point. The derivative measures the function's rate of change, essentially how "steep" the curve is at that moment.

• The process of differentiation involves finding the derivative of a function. This derivative function gives us the slope of the tangent line to the curve at any point.
• In our example, the derivative is given as \(f'(t) = t^2 - 2t + 3\).
• By calculating this derivative, we can determine how the function \(f\) behaves when its input \(t\) changes slightly.

Differentiation helps solve problems involving motion, growth, and optimization. It's a cornerstone of calculus and is widely used in fields such as physics, engineering, and economics. By understanding the derivative, we gain insights into the behavior of the function which sets the stage for finding tangent lines and integrating to find the original function.

Integration

Integration is the reverse process of differentiation. While differentiation finds the rate of change, integration determines the total accumulation of quantity or area under a curve. It is used to find functions when their derivatives are known and is often seen as "putting pieces together" to form a whole picture.

• When given a derivative, integration allows us to find the original function. In simpler terms, while differentiation breaks a function into its rate parts, integration combines them back into the complete function.
• Applying integration to the problem, we solve \(\int (t^2 - 2t + 3) dt\) to get \(f(t) = \frac{1}{3}t^3 - t^2 + 3t + C\), where \(C\) is the constant of integration.
• The constant \(C\) is determined by additional information, like a specific point the function passes through, to give us a complete solution.

Integration is a powerful tool for solving real-world problems where accumulation, area, or reverse calculation is needed. It's vital in fields such as physics, where it helps calculate trajectories and areas under velocity-time graphs.

Tangent Line

A tangent line is a straight line that just "touches" a curve at a point, without crossing it at that immediate vicinity. It represents the instantaneous direction of the curve at that point and plays a crucial role in calculus.

• The slope of this tangent line is given by the derivative of the curve at the point of tangency.
• In our exercise, knowing \(f'(t) = t^2 - 2t + 3\) tells us the slope of the tangent line at any point \(t\) on the curve of \(f(t)\).
• The tangent line becomes particularly important when examining instantaneous rates like speed, or when solving for equations that describe how systems change at an exact point.

Understanding the concept of tangent lines helps in visualizing how functions behave at specific points and translates into insights on how to control, optimize, or manipulate systems based on this behavior.