Section 2.1: The Tangent and Velocity Problems
1. The point \(P(2,-1)\) lies on the curve \(y=1 /(1-x)\).
(a) If \(Q\) is the point \((x, 1 /(1-x))\), use your calculator to find the slope of the secant line \(P Q\) (correct to six decimal places)
for the following values of \(x\) :
(i) 1.5 (ii) 1.9 (iii) 1.99 (iv) 1.999 (v) 2.5 (vi) 2.1 (vii) 2.01 (viii) 2.001
(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at \(P(2,-1)\).
(c) Using the slope from part (b), fnd an equation of the tangent line to the curve at \(P(2,-1)\).
2. If a rock is thrown upward on the planet Mars with a velocity of \(10 \mathrm{~m} / \mathrm{s}\), its height in meters \(t\) seconds later is given by \(y=10 t-1.86 t^2\).
(a) Find the average velocity over the given time intervals:
(i) \([1,2]\) (ii) \([1,1.5]\) (iii) \([1,1.1]\) (iv) \([1,1.01]\) (v) \([1,1.001]\)
(b) Estimate the instantaneous velocity when \(t=1\).