Question

Finding an Equation of a Tangent Linefind an equation of the tangent line to the graph of the function at the given point,use a graphing utility to graph the function and its tangent line at the point, anduse the tangent feature of a graphing utility to confirm your results.

Answer 1

To find the equation of the tangent line, calculate the slope at the given point by first finding the derivative of the function and then using the point-slope form with the coordinates of the point.

Step-by-step explanation:

To find the equation of the tangent line to the graph of a function at a given point, we can use differential calculus. For the given problem scenario, we are asked to find the tangent line at the point where t = 25 seconds. We know that the slope of the curve at a point is the same as the slope of the tangent line at that point, which means we ’ll need to calculate the derivative of the function to find this slope.

The first step is to identify the endpoints of the tangent line corresponding to the given point, as given in the question (1300 m at 19 s and 3120 m at 32 s). By calculating the slope v between these two points, we would obtain the slope of the tangent line.

Once the slope is known, using the point-slope form of a linear equation (y - y1 = m(x - x1)), we can substitute the slope we found and the coordinates of the point at t = 25 s to get the equation of the tangent line. To confirm our findings, we can use a graphing utility to graph the function and its tangent line, and employ the tangent feature of the utility to check if the slope and the equation of the tangent line are correct.

Answer 2

step 1

we have the function

`y=(x-2)(x^2+3x)`

Applying the distributive property

`\begin{gathered} y=x^3+3x^2-2x^2-6x \\ y=x^3+x^2-6x \end{gathered}`

Find out the first derivative of the given function y

`y^(\prime)=3x^2+2x-6`

Remember that the derivative of the function is the same that the slope

so

`m=3x^2+2x-6`

Evaluate at the point (1,-4)

`\begin{gathered} m=3(1)^2+2(1)-6 \\ m=3+2-6 \\ m=-1 \end{gathered}`

step 2

Find out the equation of the line in slope-intercept form

y=mx+b

we have

m=-1

point (1,-4)

substitute and solve for b

-4=-1(1)+b

b=-4+1

b=-3

therefore

The equation of the line is

y=-x-3

Verify with a graphing tool