Math Problem Statement
Let f(x) = -x^2 - 2x + 3. a) Find the derivative f'(x), b) Find the slope of the tangent line at (0, 3), c) Find the rate of change when x = 0, d) Find the equation of the tangent line at (0, 3), e) Sketch the graph of f and the tangent line at (0, 3).
Solution
Let's work through each part of the problem based on the given function .
a. Find the derivative , using the definition of the derivative.
The derivative of can be found by using standard differentiation rules rather than the limit definition directly:
= -2x - 2.$$ ### b. Find the slope of the tangent line to the graph of $$f$$ at the point $$(0, 3)$$. The slope of the tangent line at a specific point is the value of the derivative at that point. We already found that $$f'(x) = -2x - 2$$. To find the slope at $$x = 0$$: $$f'(0) = -2(0) - 2 = -2.$$ Thus, the slope of the tangent line at $$(0, 3)$$ is $$-2$$. ### c. Find the rate of change of $$f$$ when $$x = 0$$. This is the same as finding the derivative at $$x = 0$$, which we already did in part (b). The rate of change at $$x = 0$$ is $$-2$$. ### d. Find an equation of the tangent line to the graph of $$f$$ at the point $$(0, 3)$$. The equation of the tangent line is given by the point-slope form of a line: $$y - y_1 = m(x - x_1),$$ where $$m$$ is the slope and $$(x_1, y_1)$$ is the point on the curve. We know the slope is $$-2$$ and the point is $$(0, 3)$$, so: $$y - 3 = -2(x - 0),$$ which simplifies to: $$y = -2x + 3.$$ This is the equation of the tangent line. ### e. Sketch the graph of $$f$$ and the tangent line to the curve at the point $$(0, 3)$$. The function $$f(x) = -x^2 - 2x + 3$$ is a downward-opening parabola (since the coefficient of $$x^2$$ is negative), and the tangent line has equation $$y = -2x + 3$$, which touches the parabola at the point $$(0, 3)$$. Would you like a visual sketch of this graph and the tangent line? --- Here are 5 follow-up questions for further practice: 1. What is the second derivative $$f''(x)$$ of the function? 2. Find the vertex of the parabola $$f(x) = -x^2 - 2x + 3$$. 3. For which values of $$x$$ is the function increasing or decreasing? 4. How does the concavity of the graph of $$f$$ relate to the second derivative? 5. What are the x-intercepts of the graph of $$f(x)$$? **Tip**: The slope of the tangent line to a curve at any point gives the instantaneous rate of change of the function at that point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Tangent Lines
Quadratic Functions
Formulas
Derivative of f(x) = -2x - 2
Point-slope form of a line: y - y1 = m(x - x1)
Theorems
Derivative as the slope of the tangent line
Suitable Grade Level
Grades 10-12
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