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5 Imitate the example in the text to do each of these falling body problems (a) (b) A ball is dropped from a height of 100 feet How long will it take that ball to hit the ground Suppose that the ball from part (a) is thrown straight down with an initial velocity of 10 feet per second Then how long will it take the ball to hit the ground Suppose that the ball from part (a) is thrown straight up with an initial velocity of 10 feet per second Then how long will it take the ball to hit the ground d sin(ln(cos x)) dx d sin(cos x) e dx d ln(esin x + x) dx d arcsin(x 2 + tan x) dx d arccos(ln x ex /5) dx d arctan(x 2 + ex ) dx
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6 Use the Chain Rule to perform each of these differentiations: (a) (b) (c) (d) (e) (f)
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7 If a car has position p(t) = 6t 2 5t + 20 feet, where t is measured in seconds, then what is the velocity of that car at time t = 4 What is the average velocity of that car from t = 2 to t = 8 What is the greatest velocity over the time interval [5, 10] 8 In each of these problems, use the formula for the derivative of an inverse function to nd [f 1 ] (1) (a) (b) (c) (d) f (0) = 1, f f (3) = 1, f f (2) = 1, f f (1) = 1, f (0) = 3 (3) = 8 (2) = 2 (1) = 40
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We know that the value of the derivative of a function f at a point x represents the slope of the tangent line to the graph of f at the point (x, f (x)) If that slope is positive, then the tangent line rises as x increases from left to right, hence so does the curve (we say that the function is increasing) If instead the slope of the tangent line is negative, then the tangent line falls as x increases from left to right, hence so does the curve (we say that the function is decreasing) We summarize: On an interval where f > 0 the graph of f goes uphill On an interval where f < 0 the graph of f goes downhill See Fig 31 With some additional thought, we can also get useful information from the second derivative If f = (f ) > 0 at a point, then f is increasing Hence the slope of the tangent line is getting ever greater (the graph is concave up) The picture must be as in Fig 32(a) or 32(b) If instead f = (f ) < 0 at a point then f is decreasing Hence the slope of the tangent line is getting ever less (the graph is concave down) The picture must be as in Fig 33(a) or 33(b) Using information about the rst and second derivatives, we can render rather accurate graphs of functions We now illustrate with some examples EXAMPLE 31
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Sketch the graph of f (x) = x 2
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CHAPTER 3 Applications of the Derivative
Fig 31
Fig 32
Fig 33
SOLUTION Of course this is a simple and familiar function, and you know that its graph is a parabola But it is satisfying to see calculus con rm the shape of the graph Let us see how this works
CHAPTER 3 Applications of the Derivative
First observe that f (x) = 2x We see that f < 0 when x < 0 and f > 0 when x > 0 So the graph is decreasing on the negative real axis and the graph is increasing on the positive real axis Next observe that f (x) = 2 Thus f > 0 at all points Thus the graph is concave up everywhere Finally note that the graph passes through the origin We summarize this information in the graph shown in Fig 34
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