Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.
c. What is the height of the stone at the highest point?

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=(x²−1)sin^−1 x on [−1,1]

Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. <IMAGE>

c. Find the average velocity of the car over the interval [1.75, 2.25]. Estimate the velocity of the car at 11:00 A.M. and determine the direction in which the patrol car is moving.

Deriving trigonometric identities

c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.

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c. $p^{\prime }\left(4\right)$

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

c. d/dx ((f(x)g(x)) |x=3

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.