Suppose that we don\'t have a formula for g(x) but we know that g(1) = -3 and g\

Question

Suppose that we don't have a formula for g(x) but we know that g(1) = -3 and g'(x) = Squareroot x^2 + 15 for all x. Use a linear approximation to estimate g(0.99) and g(1.01). Are your estimates in part (a) too large or too small? Explain. The slopes of the tangent lines are positive but the tangents are becoming less steep, so the tangent lines lie below the curve. Thus, the estimates are too small. The slopes of the tangent lines are positive and the tangents are getting steeper, so the tangent lines lie below the curve. Thus, the estimates are too small. The slopes of the tangent lines are positive but the tangents are becoming less steep, so the tangent lines lie above the curve. Thus, the estimates are too large. The slopes of the tangent lines are positive and the tangents are getting steeper, so the tangent lines lie above the curve. Thus, the estimates are too large.

Explanation / Answer

a)

L(x) = g(a) + g'(a) (x-a)
for 0.99:
x= 0.99
a = 1
g(1) = -3
g'(1) = sqrt(1+15) = 4

L(x) = g(a) + g'(a) (x-a)
L(0.99) = g(1) + g'(1) (0.99-1)
= (-3) + (4)*(0.99-1)
= -3.04

L(x) = g(a) + g'(a) (x-a)
L(1.01) = g(1) + g'(1) (1.01-1)
= (-3) + (4)*(1.01-1)
= -2.96

b)
since g'(1) is positive, slope is positive
curve is going up
so tangent will be below the curve
Hence the estimate is too small
Answer: option 1

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.