Full solution

Q. The function

f

is defined by

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

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

f

when

x=-1.5

. You should round all decimals to

3

places.

\newline

Answer:

Calculate Derivative: To find the equation of the tangent line at a specific point, we need to calculate the derivative of the function to find the slope of the tangent line at that point. The derivative of $f(x)$ with respect to $x$ is $f'(x)$ .
Evaluate Derivative at $x=-1.5$ : Using the chain rule and the product rule, the derivative of $f(x) = x^2 + 3\cos(x^2 + 3x)$ is $f'(x) = 2x - 3\sin(x^2 + 3x)(2x + 3)$ .
Find y-coordinate at x= $-1$ . $5$ : Now we need to evaluate the derivative at $x = -1.5$ to find the slope of the tangent line at that point. We plug $x = -1.5$ into the derivative to get $f'(-1.5) = 2(-1.5) - 3\sin((-1.5)^2 + 3(-1.5))(2(-1.5) + 3)$ .
Calculate Slope and Point: Using a calculator, we find that $f'(-1.5) \approx -3 - 3\sin(2.25 - 4.5)(-3 + 3) \approx -3 - 3\sin(-2.25)(0) \approx -3 - 0 = -3$ . This is the slope of the tangent line at $x = -1.5$ .
Write Equation of Tangent Line: Next, we need to find the $y$ -coordinate of the point on the graph of $f(x)$ where $x = -1.5$ . We plug $x = -1.5$ into the original function to get $f(-1.5) = (-1.5)^2 + 3\cos((-1.5)^2 + 3(-1.5))$ .
Simplify Equation: Using a calculator, we find that $f(-1.5) \approx 2.25 + 3\cos(2.25 - 4.5) \approx 2.25 + 3\cos(-2.25) \approx 2.25 + 3(0.776) \approx 2.25 + 2.328 \approx 4.578$ .
Simplify Equation: Using a calculator, we find that $f(-1.5) \approx 2.25 + 3\cos(2.25 - 4.5) \approx 2.25 + 3\cos(-2.25) \approx 2.25 + 3(0.776) \approx 2.25 + 2.328 \approx 4.578$ .Now we have the slope of the tangent line, $m = -3$ , and a point on the tangent line, $(-1.5, 4.578)$ . We can use the point-slope form of a line to write the equation of the tangent line: $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is the point on the line.
Simplify Equation: Using a calculator, we find that $f(-1.5) \approx 2.25 + 3\cos(2.25 - 4.5) \approx 2.25 + 3\cos(-2.25) \approx 2.25 + 3(0.776) \approx 2.25 + 2.328 \approx 4.578$ .Now we have the slope of the tangent line, $m = -3$ , and a point on the tangent line, $(-1.5, 4.578)$ . We can use the point-slope form of a line to write the equation of the tangent line: $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is the point on the line.Plugging in the values, we get the equation of the tangent line: $y - 4.578 = -3(x - (-1.5))$ or $y - 4.578 = -3(x + 1.5)$ .
Simplify Equation: Using a calculator, we find that $f(-1.5) \approx 2.25 + 3\cos(2.25 - 4.5) \approx 2.25 + 3\cos(-2.25) \approx 2.25 + 3(0.776) \approx 2.25 + 2.328 \approx 4.578$ . Now we have the slope of the tangent line, $m = -3$ , and a point on the tangent line, $(-1.5, 4.578)$ . We can use the point-slope form of a line to write the equation of the tangent line: $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is the point on the line. Plugging in the values, we get the equation of the tangent line: $y - 4.578 = -3(x - (-1.5))$ or $y - 4.578 = -3(x + 1.5)$ . Simplifying the equation, we get $y = -3x - 4.5 + 4.578$ . Combining like terms, we get $y = -3x + 0.078$ .