Example Question - slope of tangent line
Here are examples of questions we've helped users solve.
Finding Slope of Tangent Line to a Function
The image shows a graph of a function f(x) = (x - 2)² + 1, and there's a point A(3, k) marked on it. It also shows a line d that is tangent to the graph at point A. You are asked to find the slope of the tangent line d at point A. To find the slope of the tangent line to the function at x = 3, we need to calculate the derivative of the function, which gives us the slope of the tangent line at any point x. Let's find the derivative of f(x): f(x) = (x - 2)² + 1 Taking the derivative of f(x) with respect to x: f'(x) = 2(x - 2)*1 = 2x - 4 Now, we substitute x = 3 into the derivative to find the slope of the tangent line at point A: f'(3) = 2(3) - 4 = 6 - 4 = 2 So, the slope of the tangent line at point A(3, k) is 2.
Solving for f(2) Given Tangent Line Information
The question is asking for the value of f(2) given that the tangent line to the graph of y = f(x) at the point (2, 4) passes through the point (-1, 3). To solve this, we can use the fact that the slope of the tangent line at a point on a curve is equal to the derivative of the function at that point. The slope of the tangent line can be found using the two points it passes through: (2, 4) and (-1, 3). The slope (m) of the line passing through these two points is given by: m = (y2 - y1) / (x2 - x1) Substitute the given points into the formula: m = (3 - 4) / (-1 - 2) m = (-1) / (-3) m = 1/3 Now, because the slope of the tangent line to the curve at the point (2, 4) is also the derivative of the function at x = 2, we can say that: f'(2) = 1/3 Since the point (2,4) lies on the tangent and hence on the graph of the function, we know that f(2) = 4. The question seems to be mistaken or poorly phrased because it's asking for f(2), which we already know is 4 from the given point (2, 4) on the graph of the function. Hence, f(2) is 4.
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