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6:13Highway 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>
a. Determine the average velocity of the car during the first 45 minutes of the trip.
Shrinking isosceles triangle The hypotenuse of an isosceles right triangle decreases in length at a rate of 4 m/s.
a. At what rate is the area of the triangle changing when the legs are 5 m long?
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = x2 - 4; P(2, 0)
79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)
The volume V of a sphere of radius r changes over time t.
a. Find an equation relating dV/dt to dr/dt.
{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by
f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>
Let f (x) = √x.
a. Find the exact value of f' (4).
