a. find an equation of the tangent line at x = a\nb. use a graphing utility to graph the curve and the…
a. find an equation of the tangent line at x = a\nb. use a graphing utility to graph the curve and the tangent line on the same set of axes.\ny=x³ - 3x² + 4x + 2; a = 1\na. the equation of the tangent line at a = 1 is y = x + 3.\nb. choose the correct graph below. all graphs are shown in a -3,3,1 by -20,20,5 window.
Answer
Answer:
a. The equation of the tangent line at (x = 1) is (y=x + 3) b. Without seeing the actual graphs, we can't determine the correct option. But we know the curve (y=x^{3}-3x^{2}+4x + 2) is a cubic - function and the tangent line (y=x + 3) is a straight - line. The tangent line (y=x + 3) should touch the curve (y=x^{3}-3x^{2}+4x + 2) at the point where (x = 1).
Explanation:
Step1: Find the derivative of the function
The function is (y=x^{3}-3x^{2}+4x + 2). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (y^\prime=3x^{2}-6x + 4).
Step2: Evaluate the derivative at (x = 1)
Substitute (x = 1) into (y^\prime): (y^\prime(1)=3(1)^{2}-6(1)+4=3 - 6 + 4=1). This is the slope (m) of the tangent line.
Step3: Find the (y) - coordinate when (x = 1)
Substitute (x = 1) into (y=x^{3}-3x^{2}+4x + 2): (y(1)=1^{3}-3(1)^{2}+4(1)+2=1-3 + 4+2=4).
Step4: Use the point - slope form of a line
The point - slope form is (y - y_{1}=m(x - x_{1})), where ((x_{1},y_{1})=(1,4)) and (m = 1). So (y-4=1\times(x - 1)), which simplifies to (y=x + 3).