Question

Graph the given function. Then find the slope or rate of change of the curve at the given value of \(x\), either manually, by zooming in, by using the TANGENT feature on your calculator, or numerically, as directed by your instructor. $$y=\sqrt{x}+x^{2} \quad \text { at } x=2$$

Short answer

The slope or rate of change of the function at x=2 is approximately 4.354.

Step-by-step solution

Graph the Function

Plot the function on a coordinate system. Since this function is a combination of a square root and a quadratic function, you would expect to see a parabola-like curve with a gentler slope for lower values of x (due to the square root) and then an increase in steepness for larger values of x (due to the x squared term). Use graphing software, a graphing calculator, or plot several points manually to draw the curve.

Differentiate the Function

To find the slope or rate of change at a given point, we need to differentiate the function. The derivative of y with respect to x, denoted as y', for the given function is found by differentiating each term separately: For the first term, \(\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}\)For the second term, \(\frac{d}{dx}(x^2) = 2x\)Hence, the derivative of the function is \(y' = \frac{1}{2\sqrt{x}} + 2x\).

Evaluate the Derivative at x=2

Substitute the value of x (which is 2) into the derivative to find the slope at that point: \(y'_{x=2} = \frac{1}{2\sqrt{2}} + 2(2)\)Now, calculate the value: \(y'_{x=2} = \frac{1}{2\cdot1.414} + 4\)\(y'_{x=2} = \frac{1}{2.828} + 4\) (rounded to three decimal places)\(y'_{x=2} \_approx \frac{1}{2.828} + 4\)\(y'_{x=2} \_approx 0.354 + 4\)\(y'_{x=2} \_approx 4.354\)The slope or rate of change of the curve at x=2 is approximately 4.354.

Key concepts

Differentiating Functions

When you're trying to understand the behavior of a graph at a certain point, differentiating functions is the tool you need. Differentiation, which is a fundamental concept in calculus, involves finding the derivative of a function. The derivative represents the rate at which the function's value is changing at any given point. In other words, it tells you how steep the curve is at any point.

For example, if you have a function like y = \( \sqrt{x} + x^2 \), to differentiate it, you'd need to apply the rules of differentiation to each term separately. The square root of x, which can be written as x^(1/2), has a derivative of \( \frac{1}{2}x^{-(1/2)} \) or \( \frac{1}{2\sqrt{x}} \). Similarly, the derivative of x^2 is 2x. This process allows you to find the slope of the graph at any point along the x-axis, which is crucial for understanding how the function behaves at that specific spot.

Tangent Line to a Curve

Imagine being a tiny ant walking along the curve of a function. The tangent line is like a straight path that just touches the curve at the point where you stand, without cutting through it. It's the slope of this tangent line that tells you how steeply the function is changing at that point.

The derivative you calculated for a function, y', is actually the slope of the tangent line at any x value. So, after differentiating your function, you can find the slope of the curve at a specific point by evaluating the derivative at that x value. Returning to our earlier function, y = \( \sqrt{x} + x^2 \), after you've found the derivative, you just plug in the value of x you're interested in. If x = 2, you'd substitute 2 into the derivative to get the slope at that point, thereby giving you the exact steepness of the tangent line at x = 2 on the graph.

Graphing Functions

Graphing is like painting a picture of the function's behavior across different values. It provides a visual representation of all the points that satisfy the function's equation. Graphing the function y = \( \sqrt{x} + x^2 \) would involve plotting points for various x values and then connecting these points to reveal the shape of the function.

Understanding the graph helps you visualize things like the curve's steepness and where it increases or decreases. This visual tool is powerful because it enables you to predict properties of the function without doing extensive calculations for every point. As you graph more functions, you'll start to recognize patterns like the parabolic shape of quadratic functions or the gentle incline introduced by a square root term.