The function f is twice differentiable for x > 0 with f(1) = 15 and f\"(1) = 20.

Question

The function f is twice differentiable for x > 0 with f(1) = 15 and f"(1) = 20. values of f', the derivative of f, are given for selected values of x in the table. Write an equation for the line tangent to the graph of f at x = 1. Use this line to approximate f (1.4). Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate integral^14 _1 f (x)dx. Use the approximation for integral^14 _1 f (x)dx. To estimate the value of f(1.4). Show the computations that lead to your answer.

Explanation / Answer

a) f(1)=15

f'(1)=8

equation of tangent at x = 1

y - f(1) = f'(1) (x -1)

or y - 15 = 8(x - 1)

y = 8x +7

f(1.4) using tangent is equal to 8*1.4 + 7 = 18.2

b) required integral = 0.1*(f(1) + 2 f(1.1 ) +f(1.2)+ f(1.3)+ f(1.4))

=0.1/2*(8+2*10+2*12+2*13+14.5)

=9.25/2=4.625

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