Question

Let f(x) be defined such that f(−1) = 1 and f prime of x equals cos of the quantity 1 over x squared plus x comma where −2 < x < 1.

Part A: Find the tangent line approximation for f(−0.9). (25 points)

Part B: If f(−0.9) has an actual value of 1.3, use the shape of the graph to determine if this is an overestimate or underestimate. Justify your answer. (15 points)

Answer 1

for part A:

y-1=(cos(1)-1)((-0.9)+1)

y=(-0.459)((-0.9)+1)+1

y=1.4403

f(-0.9) = 1.4403

Part B:

f''(x) = -(x^3-2)*sin(x+1/x^2) / x^3

f''(-0.9) = -1.59

The function is decreasing so therefore the estimate is an over estimate.

Step-by-step explanation:

Assuming you meant cos(1/x^2) + x with f(-1) = 1 given we can create the tangent line approximation using point slope form. Placing the given values within the equation would create y - 1 = dy/dx(x+1). We just need to sub in dy/dx which f'(x) is given which we just need to plug in -1 and then sub in -0.9 in x for the point-slope form.

For determining if the approximation is an underestimate or overestimate you can verify by checking the 2nd derivative and if it is increasing it is below and if it is decreasing it is above.