Find the equation of the line tangent to the graph of at . Use the point-slope form.
Include a multiplication sign between symbols. For example, .

Ask by Osborne Mcdonald. in the United States

Mar 13,2025

Question

Find the equation of the line tangent to the graph of $$ f(x)=3 x^{2}-7 $$ at $$ x=2 $$. Use the point-slope form. Include a multiplication sign between symbols. For example, $$ a^{*} x $$.

UpStudy AI · Accepted Answer

1. Start with the function:  
   $$
   f(x)=3 * x^{2}-7
   $$

2. Compute the derivative to find the slope of the tangent line:  
   $$
   f'(x)=6 * x
   $$

3. Evaluate the derivative at $$ x=2 $$ to get the slope:  
   $$
   f'(2)=6 * 2=12
   $$

4. Find the point on the curve at $$ x=2 $$:  
   $$
   f(2)=3 * (2)^{2}-7=3 * 4-7=12-7=5
   $$  
   So the point is $$ (2, 5) $$.

5. Use the point-slope form of the equation of a line:  
   $$
   y-y_{1}=m*(x-x_{1})
   $$  
   where $$ m=12 $$ and $$ (x_{1}, y_{1})=(2, 5) $$.

6. Substitute the values into the point-slope form:  
   $$
   y-5=12*(x-2)
   $$

Thus, the equation of the tangent line is:  
$$
y-5=12*(x-2)
$$
The equation of the tangent line is $$ y - 5 = 12(x - 2) $$.

Find the equation of the line tangent to the graph of at . Use the point-slope form.
Include a multiplication sign between symbols. For example, .

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Find the equation of the line tangent to the graph of at . Use the point-slope form. Include a multiplication sign between symbols. For example, .