Suppose the linear approximation for a function f(x) at a = 2 is given by the tangent line y = −2x + 10.What are f(2) and f '(2)?

Recall that the slope m of the graph of f at the point (x, f(x)) is equal to the slope of its tangent line at (x, f(x)), and is given by the following provided this limit exists.m = lim h→0 msec = lim h→0 f(x + h) − f(x)hTherefore, we will first need to find the expression that represents the secant line at the point (2, −10).First, set up the difference equation for the point (2, −10) by substituting x = 2.m  =  lim h→0 f(x + h) − f(x)h =  lim h→0 f  + h − f  h

f(x)=−3x 2 +ax+bFor the quadratic function given above, a and b are constants such the x-coordinate of the vertex is 2.5. If the graph of the function given above is tangential to the line y=10, then find the value of f(3).

Find the linearization of the function f(x) = x + 6 at a = 3 and use it to approximate the numbers 8.99 and 9.01. Are these approximations overestimates or underestimates?SOLUTION The derivative of f(x) = (x + 6)1/2 isf '(x) = (12​)(x+6)(−(12​))·1 .and so we have f(3) = and f '(3) = . Putting these values into this equation, we see that the linearization isL(x)= f(3) + f '(3)(x − 3) = + (x − 3)= (16​)x+172​ .The corresponding linear approximation isx + 6 ≈ + x6        (when x is near 3).In particular, we have8.99 ≈ 52 + 6 =     (round to four decimal places)and  9.01 ≈ 52 + 6 =     (round to four decimal places).The linear approximation is illustrated in the figure to the left. We see that, indeed, the tangent line approximation is a good approximation to the given function when x is near 3. We also see that our approximations are overestimates because the tangent line lies above the curve.     Of course, a calculator could give us approximations for 8.99 and 9.01, but the linear approximation gives an approximation over an entire interval.

The corresponding linear approximation isx + 6 ≈ + x6        (when x is near 3).In particular, we have8.99 ≈ 52 + 6 =     (round to four decimal places)and  9.01 ≈ 52 + 6 =     (round to four decimal places).The linear approximation is illustrated in the figure to the left. We see that, indeed, the tangent line approximation is a good approximation to the given function when x is near 3. We also see that our approximations are overestimates because the tangent line lies above the curve.     Of course, a calculator could give us approximations for 8.99 and 9.01, but the linear approximation gives an approximation over an entire interval.

Find the slope of f(x)𝑓(𝑥) at x=2𝑥=2. The graph of f(x)𝑓(𝑥) is shown below.Move the point on the curve to x=2𝑥=2. Then plot two points on the tangent line. Finally, calculate the slope of f(x)𝑓(𝑥) at x=2𝑥=2. Write your answer as a simplified fraction or rounded to 44 decimal places.