Given that f(-0.5)=2 and f'(-0.5)=4, use a tangent line approximation to estimate f(0).
The lesson is not very clear on what a tangent line approximaton is.
The tangent line approximation is:
f(x) = f(a) + (x-a)*f'(a)
It is the formula for a straight line that goes through x = a with the slope of f(x) at x=a.
Let a = -0.5 and x = 0
f(0) = 2 + (0.5)*4 = 4
Ah, the thrilling world of tangent line approximations! Allow me to explain in my own quirky way.
Imagine you're on a roller coaster, and at one point, the ride suddenly takes a sharp turn. You might lean to the side, right? Well, think of the tangent line as the "leaning" line. It's the line that just touches the graph at a specific point, kind of like how you would lean against a wall while waiting for your turn on the roller coaster.
In this case, we know that f(-0.5) = 2, so we have a point on the graph (-0.5, 2). And we know that f'(-0.5) = 4, which is the slope of the tangent line at that point.
Now, let's use this info to estimate f(0). Since we're using a tangent line approximation, we'll pretend that the graph is a straight line near the point (-0.5, 2). We'll use the slope of the tangent line (4) to sketch that straight line.
The fun part is that the tangent line is (almost) the same as the graph of the function near that point. So, to estimate f(0), we just need to find where our straight line intersects the x-axis.
Hope that clears things up a bit! Keep riding the tangent line roller coaster of math!
A tangent line approximation is a method used to estimate the value of a function at a point by using the slope of the tangent line at another point. The basic idea behind this method is that if the function is approximately linear in a small interval around the point of interest, then the tangent line at a nearby point can be used to approximate the function.
To estimate f(0) using a tangent line approximation, we can start by finding the equation of the tangent line at the point (-0.5, 2) using the given information.
Step 1: Find the equation of the tangent line:
Since the slope of the tangent line is given by f'(-0.5) = 4, we can write the equation of the tangent line as:
y - f(-0.5) = f'(-0.5)(x - (-0.5))
Simplifying this equation, we get:
y - 2 = 4(x + 0.5)
Step 2: Evaluate f(0) using the tangent line approximation:
To estimate f(0), we substitute x = 0 into the equation of the tangent line:
y - 2 = 4(0 + 0.5)
Simplifying this equation further, we get:
y - 2 = 4(0.5)
y - 2 = 2
Adding 2 to both sides of the equation, we find:
y = 4
Therefore, the tangent line approximation estimates that f(0) is equal to 4.
A tangent line approximation is a method used to estimate the value of a function at a particular point by using the slope of the tangent line at a nearby point. It is based on the idea that a small interval around a point on a smooth curve can be approximated by a straight line.
To estimate f(0) using a tangent line approximation, we need to find the equation of the tangent line at x = -0.5 and then use that equation to determine the value of f(0).
Step 1: Determine the equation of the tangent line
The equation of a line can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. To find the slope, we use the derivative f'(-0.5) = 4, which gives us the slope of the tangent line at x = -0.5.
Step 2: Find the y-coordinate (f(0)) using the equation of the line
To estimate f(0), we need to determine the value of y when x = 0. We can substitute x = 0 into the equation of the line obtained in step 1 to find this value.
Let's put these steps into action:
Step 1: The slope of the tangent line at x = -0.5 is given as f'(-0.5) = 4.
Step 2: To find the y-coordinate (f(0)), we need to determine the value of y when x = 0. Using the equation of the line, we substitute x = 0:
y = mx + b
f(0) = 4(0) + b
f(0) = b
Therefore, the value of f(0) is equal to the y-intercept b.
In this case, since we don't have the value of b, we cannot determine the exact value of f(0) using the information provided.
However, the tangent line approximation provides us with an estimate for f(0) based on the assumption that the function is approximately linear in a small neighborhood around x = -0.5.