Carson M.

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

(a) $ y = \pi^x $
(b) $ y = x^\pi $
(c) $ y = x^2 (2 - x^3) $
(d) $ y = \tan t - \cos t $
(e) $ y = \frac{s}{1 + s} $
(f) $ y = \frac{\sqrt {x^3 - 1}}{1 + \sqrt[3]{x}} $

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 $ PQ $ (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), find an equation of the tangent line to the curve at $ P(2, -1) $.

The point $ P(0.5, 0) $ lies on the curve $ y = \cos \pi x $.

(a) If $ Q $ is the point $ (x, \cos \pi x) $, use your calculator to find the slope of the secant line $ PQ $ (correct to six decimal places) for the following values of $ x $:

(i) $ 0 $ (ii) $ 0.4 $ (iii) $ 0.49 $
(iv) $ 0.499 $ (v) $ 1 $ (vi) $ 0.6 $
(vii) $ 0. 51 $ (viii) $ 0.501 $

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $ P(0.5, 0) $.

(c) Using the slope from part (b), find an equation of the tangent line to the curve at $ P(0.5, 0) $.

(d) Sketch the curve, two of the secant lines, and the tangent line.

If a ball is thrown into the air with a velocity of $ 40 ft/s $, its height in feet $ t $ seconds later is given by $ y = 40t - 16t^2 $.

(a) Find the average velocity for the time period beginning when $ t = 2 $ and lasting
(i) 0.5 seconds (ii) 0.1 seconds
(iii) 0.05 seconds (iv) 0.01 seconds
(b) Estimate the instantaneous velocity when $ t = 2 $.

If a rock is thrown upward on the planet Mars with a velocity of $ 10 m/s $, its height in meters $ t $ seconds after is given by $ y = 10t - 1.86t^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 $.

The table shows the position of a motorcyclist after accelerating from rest.

(a) Find the average velocity for each time period:
(i) $ [2, 4] $ (ii) $ [3, 4] $ (iii) $ [4, 5] $ (iv) $ [4, 6] $
(b) Use the graph of $ s $ as a function of $ t $ to estimate the instantaneous velocity when $ t = 3 $.