Consider the graph of the parabola f(x)equals=x squaredx2. For xgreater than0 and hgreater than0, the secant line through (x,f(x)) and (xplus+h,f(x+h)) always has a greater slope than the tangent line at (x,f(x)).

Question

Consider the graph of the parabola ​f(x)equals=x squaredx2. For xgreater than>0 and hgreater than>​0, the secant line through​ (x,f(x)) and ​(xplus+​h,f(x+h)) always has a greater slope than the tangent line at​ (x,f(x)).

Choose the correct answer below.
A.
The statement is true. Since the slope of x squaredx2 is increasing for xgreater than>​0, the average rate of growth on the interval ​[x,xplus+​h] is greater than the instantaneous rate of growth at x.
B.
The statement is true. The slope of the secant line through​ (x,f(x)) and ​(xplus+​h,f(xplus+​h)) is left parenthesis x plus h right parenthesis squared(x+h)2​, which is greater than x squaredx2​, the slope of the tangent line at​ (x,f(x)).
C.
The statement is false. The tangent line at​ (x,f(x)) is the limit of secant lines through​ (x,f(x)) and ​(xplus+​h,f(xplus+​h)) as h goes to zero.
D.
The statement is false. If x and h both equal​ 1, then the tangent line at point​ (1,1) and the secant line through​ (1,1) and​ (2,4) have the same slope.

Steve · Accepted Answer

while the tangent line is the limit of the secants lines, that was not the question. It said h > 0. Draw a parabola. The secants are always steeper than the tangents. So, (A)

Consider the graph of the parabola ​f(x)equals=x squaredx2. For xgreater than>0 and hgreater than>​0, the secant line through​ (x,f(x)) and ​(xplus+​h,f(x+h)) always has a greater slope than the tangent line at​ (x,f(x)).

Consider the graph of the parabola f(x)equals=x squaredx2. For xgreater than>0 and hgreater than>0, the secant line through (x,f(x)) and (xplus+h,f(x+h)) always has a greater slope than the tangent line at (x,f(x)).