Solved: ?A tank holds 1000 gallons of water, which drains from the bottom of the tank in

A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. (a) If P is the point (15, 250) on the graph of V, find the slopes of the secant lines PQ when Q is the point on the graph with t=5, 10, 20, 25, and 30 . (b) Estimate the slope of the tangent line at P by averaging the slopes of two secant lines. (c) Use a graph of V to estimate the slope of the tangent line at P. (This slope represents the rate at which the water is flowing from the tank after 15 minutes.)

A student bought a smartwatch that tracks the number of steps she walks throughout the day. The table shows the number of steps recorded \(t\) minutes after 3:00 PM on the first day she wore the watch. (a) Find the slopes of the secant lines corresponding to the given intervals of \(t\). What do these slopes represent? (i) \([0,40]\) (ii) \([10,20]\) (iii) \([20,30]\) (b) Estimate the student's walking pace, in steps per minute, at 3:20 PM by averaging the slopes of two secant lines. Equation Transcription: Text Transcription: t t [0,40] [10,20] [20,30]

The point \(P(2,-1)\) lies on the curve \(y=1 /(1-x)\). (a) If \(Q\) is the point \((x, 1 /(1-x))\), find the slope of the secant line \(P Q\) (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 \(P(2,-1)\). Equation Transcription: Text Transcription: P(2,-1) y=1/(1-x) Q (x,1/1-x) PQ x P(2,-1) P(2,-1)

The point \(P(0.5,0)\) lies on the curve \(y=\cos \pi x\). (a) If \(Q\) is the point \((x, \cos \pi x)\), 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. Equation Transcription: Text Transcription: P(0.5,0) y=cos pix Q (x,cos pix) PQ x P(0.5,0) P(0.5,0)

The deck of a bridge is suspended 275 feet above a river. If a pebble falls off the side of the bridge, the height, in feet, of the pebble above the water surface after \(t\) seconds is given by \(y=275-16 t^{2}\). (a) Find the average velocity of the pebble for the time period beginning when \(t=4\) and lasting (i) 0.1 seconds (ii) 0.05 seconds (iii) 0.01 seconds (b) Estimate the instantaneous velocity of the pebble after 4 seconds. Equation Transcription: Text Transcription: y=275-16t^2 t=4

If a rock is thrown upward on the planet Mars with a velocity of \(10 \mathrm{~m} / \mathrm{s}\), its height in meters \(t\) seconds later is given by \(y=10 t-1.86 t^{2}\). (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 \(t=1\). Equation Transcription: Text Transcription: 10 m/s t y=10t-1.86t^2 [1,2] [1,1.5] [1,1.1] [1,1.01] [1,1.001]

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 \(t=3\). Equation Transcription: Text Transcription: [2,4] [3,4] [4,5] [4,6] s t t=3

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion \(s=2 \sin \pi t+3 \cos \pi t\) , where \(t\) is measured in seconds. (a) Find the average velocity during each time period: \(\text { (i) }[1,2]\) \(\text { (ii) }[1,1.1]\) \(\text { (iii) }[1,1.01]\) \(\text { (iv) }[1,1.001]\) (b) Estimate the instantaneous velocity of the particle when \(t=1\). Equation Transcription: Text Transcription: s=2 sin pit+3 cos pit t (i)[1,2] (ii)[1,1.1] (iii)[1,1.01] (iv)[1,1.001] t=1

The point \(P(1,0)\) lies on the curve \(y=\sin (10 \pi / x)\). (a) If \(Q\) is the point \((x, \sin (10 \pi / x))\) , find the slope of the secant line \(PQ\) (correct to four decimal places) for \(x=2\), 1.5, 1.4, 1.3, 1.2, 1.1,0.5,0.6,0.7,0.8, and 0.9. Do the slopes appear to be approaching a limit? (b) Use a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tangent line at \(P\). (c) By choosing appropriate secant lines, estimate the slope of the tangent line at \(P\). Equation Transcription: Text Transcription: P(1,0) y=sin(10pi/x) Q (x,sin(10pi/x)) x=2 P P

A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. (a) If P is the point (15,250) on the graph of V, find the slopes of the secant lines PQ when Q is the point on the graph with t=5,10,20,25, and 30 . (b) Estimate the slope of the tangent line at P by averaging the slopes of two secant lines. (c) Use a graph of V to estimate the slope of the tangent line at P. (This slope represents the rate at which the water is flowing from the tank after 15 minutes.) 2. A student bought a smartwatch that tracks the number of steps she walks throughout the day. The table shows the number of steps recorded t minutes after 3:00 PM on the first day she wore the watch.

A student bought a smartwatch that tracks the number of steps she walks throughout the day. The table shows the number of steps recorded minutes after 3:00 on the first day she wore the watch. 0 10 20 30 40 Steps 3438 4559 5622 6536 7398 (a) Find the slopes of the secant lines corresponding to the given intervals of . What do these slopes represent? (i) (ii) (iii) (b) Estimate the student's walking pace, in steps per minute, at 3:20 PM by averaging the slopes of two secant lines.

The point lies on the curve . (a) If is the point , find the slope of the secant line (correct to six decimal places) for the following values of : (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at . (c) Using the slope from part (b), find an equation of the tangent line to the curve at