Given: y=−2x2−5x+7, the equation of the tangent line at the point with abscissa 2 is...
Explanation
Given the quadratic function y=−2x2−5x+7. We need to find the equation of the tangent line at the point with abscissa x1=2.
First, we find the ordinate (y1) of the tangent point by substituting x1=2 into the curve equation:
y1=−2(2)2−5(2)+7
y1=−2(4)−10+7
y1=−8−10+7
y1=−11
So, the tangent point is (2,−11).
Second, we find the gradient of the tangent line (m) using the first derivative (y′):
y′=−4x−5
Substitute x=2 into the first derivative to get the gradient:
m=−4(2)−5
m=−8−5
m=−13
Third, we determine the tangent line equation using the formula y−y1=m(x−x1):
y−(−11)=−13(x−2)
y+11=−13x+26
Move all terms to the left side to get the form Ax+By+C=0:
13x+y+11−26=0
13x+y−15=0
So, the equation of the tangent line is 13x+y−15=0.