The function $f$ is defined by $f(x)=x^{2}+3 x+4 \cos (3 x)$ . Use a calculator to write the equation of the line tangent to the graph of $f$ when $x=2.5$ . You should round all decimals to $3$ places. $\newline$ Answer:

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Q. The function

f

is defined by

f(x)=x^{2}+3 x+4 \cos (3 x)

. Use a calculator to write the equation of the line tangent to the graph of

f

when

x=2.5

. You should round all decimals to

3

places.

\newline

Answer:

Calculate Derivative: To find the equation of the tangent line at $x = 2.5$ , we first need to calculate the derivative of the function $f(x) = x^2 + 3x + 4\cos(3x)$ , which will give us the slope of the tangent line at that point.
Evaluate Derivative at $x=2.5$ : The derivative of $f(x)$ with respect to $x$ is $f'(x) = 2x + 3 - 12\sin(3x)$ (using the power rule for $x^2$ and $3x$ , and the chain rule for $4\cos(3x)$ ).
Find Slope of Tangent Line: Now we evaluate the derivative at $x = 2.5$ to find the slope of the tangent line: $f'(2.5) = 2(2.5) + 3 - 12\sin(3\times2.5)$ . Using a calculator, we find $f'(2.5) \approx 5 + 3 - 12\sin(7.5) \approx 8 - 12\sin(7.5)$ .
Find y-coordinate at $x=2.5$ : Using a calculator to find the value of $\sin(7.5)$ and then multiplying by $-12$ , we get: $-12\sin(7.5) \approx -12 \times (-0.657) \approx 7.884$ .
Use Point-Slope Form: Adding this to $5 + 3$ , we get the slope of the tangent line: $m = 8 + 7.884 \approx 15.884$ .
Convert to Slope-Intercept Form: Next, we need to find the $y$ -coordinate of the point on the function where $x = 2.5$ . We do this by evaluating $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ . Using a calculator, we find $f(2.5) \approx 6.25 + 7.5 + 4\cos(7.5) \approx 13.75 + 4\cos(7.5)$ .
Convert to Slope-Intercept Form: Next, we need to find the $y$ -coordinate of the point on the function where $x = 2.5$ . We do this by evaluating $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ . Using a calculator, we find $f(2.5) \approx 6.25 + 7.5 + 4\cos(7.5) \approx 13.75 + 4\cos(7.5)$ .Using a calculator to find the value of $\cos(7.5)$ and then multiplying by $4$ , we get: $4\cos(7.5) \approx 4 \cdot 0.755 \approx 3.020$ .
Convert to Slope-Intercept Form: Next, we need to find the $y$ -coordinate of the point on the function where $x = 2.5$ . We do this by evaluating $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ . Using a calculator, we find $f(2.5) \approx 6.25 + 7.5 + 4\cos(7.5) \approx 13.75 + 4\cos(7.5)$ .Using a calculator to find the value of $\cos(7.5)$ and then multiplying by $4$ , we get: $4\cos(7.5) \approx 4 \cdot 0.755 \approx 3.020$ .Adding this to $13.75$ , we get the $y$ -coordinate of the point on the function: $y = 13.75 + 3.020 \approx 16.770$ .
Convert to Slope-Intercept Form: Next, we need to find the $y$ -coordinate of the point on the function where $x = 2.5$ . We do this by evaluating $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ . Using a calculator, we find $f(2.5) \approx 6.25 + 7.5 + 4\cos(7.5) \approx 13.75 + 4\cos(7.5)$ .Using a calculator to find the value of $\cos(7.5)$ and then multiplying by $4$ , we get: $4\cos(7.5) \approx 4 \cdot 0.755 \approx 3.020$ .Adding this to $13.75$ , we get the $y$ -coordinate of the point on the function: $y = 13.75 + 3.020 \approx 16.770$ .Now we have the slope of the tangent line ( $x = 2.5$ $0$ ) and a point on the tangent line ( $x = 2.5$ , $x = 2.5$ $2$ ). We can use the point-slope form of a line to write the equation of the tangent line: $x = 2.5$ $3$ , where $x = 2.5$ $4$ is the point on the line.
Convert to Slope-Intercept Form: Next, we need to find the $y$ -coordinate of the point on the function where $x = 2.5$ . We do this by evaluating $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ . Using a calculator, we find $f(2.5) \approx 6.25 + 7.5 + 4\cos(7.5) \approx 13.75 + 4\cos(7.5)$ .Using a calculator to find the value of $\cos(7.5)$ and then multiplying by $4$ , we get: $4\cos(7.5) \approx 4 \cdot 0.755 \approx 3.020$ .Adding this to $13.75$ , we get the $y$ -coordinate of the point on the function: $y = 13.75 + 3.020 \approx 16.770$ .Now we have the slope of the tangent line ( $x = 2.5$ $0$ ) and a point on the tangent line ( $x = 2.5$ , $x = 2.5$ $2$ ). We can use the point-slope form of a line to write the equation of the tangent line: $x = 2.5$ $3$ , where $x = 2.5$ $4$ is the point on the line.Substituting the values into the point-slope form, we get: $x = 2.5$ $5$ . This is the equation of the tangent line in point-slope form.
Convert to Slope-Intercept Form: Next, we need to find the $y$ -coordinate of the point on the function where $x = 2.5$ . We do this by evaluating $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ . Using a calculator, we find $f(2.5) \approx 6.25 + 7.5 + 4\cos(7.5) \approx 13.75 + 4\cos(7.5)$ .Using a calculator to find the value of $\cos(7.5)$ and then multiplying by $4$ , we get: $4\cos(7.5) \approx 4 \cdot 0.755 \approx 3.020$ .Adding this to $13$ . $75$ , we get the $y$ -coordinate of the point on the function: $y = 13.75 + 3.020 \approx 16.770$ .Now we have the slope of the tangent line ( $m = 15.884$ ) and a point on the tangent line ( $x = 2.5$ , $x = 2.5$ $0$ ). We can use the point-slope form of a line to write the equation of the tangent line: $x = 2.5$ $1$ , where $x = 2.5$ $2$ is the point on the line.Substituting the values into the point-slope form, we get: $x = 2.5$ $3$ . This is the equation of the tangent line in point-slope form.To write the equation in slope-intercept form ( $x = 2.5$ $4$ ), we need to solve for $y$ : $x = 2.5$ $6$ .
Convert to Slope-Intercept Form: Next, we need to find the y-coordinate of the point on the function where $x = 2.5$ . We do this by evaluating $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ . Using a calculator, we find $f(2.5) \approx 6.25 + 7.5 + 4\cos(7.5) \approx 13.75 + 4\cos(7.5)$ . Using a calculator to find the value of $\cos(7.5)$ and then multiplying by $4$ , we get: $4\cos(7.5) \approx 4 \cdot 0.755 \approx 3.020$ . Adding this to $13.75$ , we get the y-coordinate of the point on the function: $y = 13.75 + 3.020 \approx 16.770$ . Now we have the slope of the tangent line ( $m = 15.884$ ) and a point on the tangent line ( $x = 2.5$ , $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $0$ ). We can use the point-slope form of a line to write the equation of the tangent line: $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $1$ , where $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $2$ is the point on the line. Substituting the values into the point-slope form, we get: $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $3$ . This is the equation of the tangent line in point-slope form. To write the equation in slope-intercept form ( $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $4$ ), we need to solve for $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $5$ : $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $6$ . Calculating the value of $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $7$ using a calculator, we get: $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $8$ .
Convert to Slope-Intercept Form: Next, we need to find the $y$ -coordinate of the point on the function where $x = 2.5$ . We do this by evaluating $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ . Using a calculator, we find $f(2.5) \approx 6.25 + 7.5 + 4\cos(7.5) \approx 13.75 + 4\cos(7.5)$ . Using a calculator to find the value of $\cos(7.5)$ and then multiplying by $4$ , we get: $4\cos(7.5) \approx 4 \cdot 0.755 \approx 3.020$ . Adding this to $13.75$ , we get the $y$ -coordinate of the point on the function: $y = 13.75 + 3.020 \approx 16.770$ . Now we have the slope of the tangent line ( $x = 2.5$ $0$ ) and a point on the tangent line ( $x = 2.5$ , $x = 2.5$ $2$ ). We can use the point-slope form of a line to write the equation of the tangent line: $x = 2.5$ $3$ , where $x = 2.5$ $4$ is the point on the line. Substituting the values into the point-slope form, we get: $x = 2.5$ $5$ . This is the equation of the tangent line in point-slope form. To write the equation in slope-intercept form ( $x = 2.5$ $6$ ), we need to solve for $y$ : $x = 2.5$ $8$ . Calculating the value of $x = 2.5$ $9$ using a calculator, we get: $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $0$ . Finally, we have the equation of the tangent line in slope-intercept form: $f(2.5) = (2.5)^2 + 3(2.5) + 4\cos(3\cdot2.5)$ $1$ .