1. Find the slope m and an equation of the tangent line to the graph of the function f a the point (2, 38)
2. Find an equation of the line that passes through the point (9,7) and is perpendicular to the line 5x+3y-4=0
1. "the function" is not specified.
2.
The equation perpendicular to the line
L: 5x+3y-4=0
is
L1 : 3x-5y+k=0
where k is a constant to be determined.
The equation of L1 is obtained by interchanging the coefficients of x and y, and changing the sign of x or y.
The constant k can be determined by the fact that L1 passes through (9,7).
3(9)-5(7)+k=0
k=35-27=8
So
L1: 3x-5y+8=0
1. To find the slope (m) and equation of the tangent line to the graph of the function f at the point (2, 38), follow these steps:
Step 1: Determine the derivative of the function f(x) with respect to x. Let's denote the derivative as f'(x).
Step 2: Substitute the x-coordinate of the given point into the derivative f'(x) to find the slope at that point. In this case, substitute x = 2 into f'(x) to obtain the slope m.
Step 3: Once you have the slope (m), you can use the point-slope form of a line to find the equation of the tangent line. Since you have the point (2, 38), you can use the point-slope form: y - y1 = m(x - x1), where x1 = 2, y1 = 38, and m is the slope obtained from Step 2.
Substituting the values into the equation, you can simplify it to get the equation of the tangent line.
2. To find an equation of the line that passes through the point (9, 7) and is perpendicular to the line 5x + 3y - 4 = 0, follow these steps:
Step 1: Rewrite the given equation in slope-intercept form (y = mx + b) by solving it for y. In this case, rearrange the equation 5x + 3y - 4 = 0 to get y = (-5/3)x + 4/3.
Step 2: Recall that the slope of a line perpendicular to a given line is the negative reciprocal of the original slope. In this case, the original slope is -5/3, so the slope of the perpendicular line is 3/5.
Step 3: Use the point-slope form of a line with the given point (9, 7) and the perpendicular slope of 3/5. The point-slope form is given by y - y1 = m(x - x1), where x1 = 9, y1 = 7, and m is the perpendicular slope.
Substituting the values into the equation, you can simplify it to obtain the equation of the line perpendicular to the given line that passes through the point (9, 7).