Find the slope of the tangent line for the following functions at the specified points: 1. y = 5x + 3 at point x=2 2. y = 5x^2 + 5x at point x 3. y = 8x^2 at point x
Understand the Problem
The question asks us to find the slope of the tangent line for several equations at specified points. This involves finding the derivative for each function and then evaluating the derivative at the given x-value to find the slope of the tangent line at that point.
Answer
The slope of the tangent line for $y = 5x + 3$ at $x = 2$ is $5$. The derivative of $y = 5x^2 + 5x$ is $\frac{dy}{dx} = 10x + 5$. The derivative of $y = 8x^2$ : is $\frac{dy}{dx} = 16x$.
Answer for screen readers
The slope of the tangent line for $y = 5x + 3$ at $x = 2$ is 5. The derivative of $y = 5x^2 + 5x$ is $\frac{dy}{dx} = 10x + 5$. The derivative of $y = 8x^2$ : is $\frac{dy}{dx} = 16x$.
Steps to Solve
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Find the derivative of the first equation Given $y = 5x + 3$, we find the derivative with respect to $x$. $$ \frac{dy}{dx} = 5 $$
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Evaluate the derivative at $x = 2$ Since the derivative is a constant, the slope of the tangent line at $x=2$ is simply 5.
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Find the derivative of the second equation Given $y = 5x^2 + 5x$, we find the derivative with respect to $x$. $$ \frac{dy}{dx} = 10x + 5 $$
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The problem doesn't specify where to evaluate the derivative
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Find the derivative of the third equation Given $y = 8x^2$, we find the derivative with respect to $x$. $$ \frac{dy}{dx} = 16x $$
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The problem doesn't specify where to evaluate the derivative
The slope of the tangent line for $y = 5x + 3$ at $x = 2$ is 5. The derivative of $y = 5x^2 + 5x$ is $\frac{dy}{dx} = 10x + 5$. The derivative of $y = 8x^2$ : is $\frac{dy}{dx} = 16x$.
More Information
The derivative of a function at a specific point gives the slope of the tangent line to the function at that point. For the linear function, the derivative (slope) is constant.
Tips
A common mistake is not applying the power rule correctly when finding derivatives. Another mistake is forgetting to substitute the given $x$-value into the derivative to find the slope at that specific point. For linear functions like $y = 5x+3$, students might incorrectly apply the power rule, forgetting that the derivative of $5x$ is simply 5.
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