Problem 50

Question

Given that the line $y=2 x$ is tangent to the graph of $y=x^{2}+c$, find $c .$

Step-by-Step Solution

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Answer

The value of c is 1, making the equation of the graph tangent to the line y = 2x to be y = x^2 + 1.

1Step 1: Differentiate the equation of the graph

First, differentiate the equation y = x^2 + c with respect to x. This will give us the equation for the slope of the curve at any point. \[ \frac{dy}{dx} = 2x \]

2Step 2: Find the slope of the tangent line

The given line equation is y = 2x. The slope of this line is 2. So we must find the point on the curve where the derivative is equal to the slope of the given line. Set the two equations equal to each other: \[ 2x = 2 \]

3Step 3: Solve the system of equations

Solve the equation to find the x-coordinate of the point of tangency: \[ x = 1 \] Now, find the corresponding y-coordinate by substituting the value of x back into the given equation: \[ y = 2(1) = 2 \] Thus, the point of tangency is (1, 2).

4Step 4: Determine the value of c

Now that we have the point of tangency (1, 2), we can find the value of c by substituting the x and y coordinates back into the equation y = x^2 + c. \[ 2 = 1^2 + c \] Solve for c: \[ c = 2 - 1 = 1 \] So the value of c is 1. The equation of the graph that is tangent to the line y = 2x is y = x^2 + 1.

Key Concepts

DifferentiationTangent LineQuadratic FunctionsDerivatives

Differentiation

Differentiation is an essential concept in calculus, which involves finding the rate at which a function is changing at any given point. This process is crucial in understanding how functions behave in various contexts. Differentiation helps us calculate the slope of a curve at any particular point, which can then be used to explore various physical and abstract phenomena. In simple terms, when you differentiate a function, you are finding its derivative.

When you have a function, say the quadratic function $ y = x^2 + c $, differentiation allows you to compute its derivative $ \frac{dy}{dx} $. This results in the expression $ 2x $, which tells us the slope of the function at any point $ x $.

Here are a few points to remember about differentiation:

It helps understand how a quantity changes.
The result of differentiation is the derivative, which describes the slope of the original function.

Understanding differentiation gives you deeper insights into the behavior of mathematical functions and the underlying changes they undergo over different intervals.

Tangent Line

A tangent line is a straight line that just "touches" a curve at a specific point without crossing over. It's a critical tool in calculus because it gives us valuable information about the curve's behavior at that point. For a given curve, the tangent line represents the best linear approximation at that point.

In the exercise, the equation $ y = 2x $ acts as the tangent line to the curve $ y = x^2 + c $. At the point of tangency, both the curve and the line will have the same slope. Here, the slope of the tangent line is 2.

This means:

The slope of the tangent line should match the slope of the curve $ y = x^2 + c $ at the point of contact.
At this point, the derivative of the curve should equal the slope of the tangent line.
The problem beautifully shows how you find the exact point where the tangent line meets the curve by equating their slopes.

A tangent line's role is to approximate the behavior of the curve at a specific point, enabling mathematicians to analyze and interpret the changes more smoothly.

Quadratic Functions

Quadratic functions are mathematical expressions in the form $ y = ax^2 + bx + c $, where $ a $, $ b $, and $ c $ are constants. These functions graph out into a shape known as a parabola. In our exercise, the quadratic function is $ y = x^2 + c $.

Quadratics have several interesting features:

The coefficient $ a $ determines whether the parabola opens upwards or downwards.
The constant $ c $ shifts the entire graph up or down.
They always have a vertex, which is the highest or lowest point on the graph.

The main challenge in the exercise involved finding the correct value of $ c $ such that the given line is tangent to this quadratic curve. Once $ c $ was correctly determined, the quadratic reflected this perfect tangency. This showcases how quadratic functions can be adjusted and solved for specific constraints, such as points of tangency.

Derivatives

Derivatives form the foundation of calculus, acting as indicators of how a function changes. They represent the rate at which a quantity changes, effectively functioning as a tool for understanding dynamic behavior in mathematics and real-world contexts.

In our solution, the derivative $ \frac{dy}{dx} = 2x $ provides insight into the slope of the curve $ y = x^2 + c $ at any point $ x $. This slope indicates how steep the curve is at that point, allowing us to pinpoint where the tangent line $ y = 2x $ should touch the curve.

Consider these facts about derivatives:

They help determine maximum and minimum points in a function.
They can tell you the velocity of an object when applied to motion equations.
The derivative being equal to the slope of a tangent line is a fundamental concept in solving tangent problems.

Through derivatives, you gain a powerful perspective on the ever-changing nature of functions, allowing for precise mathematical analysis and solutions.

Problem 50

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