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A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

(a) find an equation of the tangent line to the graph of \(f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(f(x)=x^{3}+1, \quad(1,2)\)

A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat (see figure). (a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock? (b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?

Find the slope of the tangent line to the graph at the given point. Bifolium: \(\left(x^{2}+y^{2}\right)^{2}=4 x^{2} y\) Point: \((1,1)\)

Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function \(\quad\) Point \(f(x)=-\frac{1}{2}+\frac{7}{5} x^{3}\) \(\left(0,-\frac{1}{2}\right)\)

A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground (see figure). When he is 10 feet from the base of the light, (a) at what rate is the tip of his shadow moving? (b) at what rate is the length of his shadow changing?

The endpoints of a movable rod of length 1 meter have coordinates \((x, 0)\) and \((0, y)\) (see figure). The position of the end on the \(x\) -axis is \(x(t)=\frac{1}{2} \sin \frac{\pi t}{6}\) where \(t\) is the time in seconds. (a) Find the time of one complete cycle of the rod. (b) What is the lowest point reached by the end of the rod on the \(y\) -axis? (c) Find the speed of the \(y\) -axis endpoint when the \(x\) -axis endpoint is \(\left(\frac{1}{4}, 0\right)\).

Find the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete cycles in the interval \([0,2 \pi] .\) What can you conclude about the slope of the sine function \(\sin a x\) at the origin? a.\(y=\sin x\) b.\(y=\sin 2 x\)

Find the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete cycles in the interval \([0,2 \pi] .\) What can you conclude about the slope of the sine function \(\sin a x\) at the origin? a.\(y=\sin 3 x\) b.\(y=\sin \frac{x}{2}\)

Sketch a graph of a function whose derivative is always negative.