Average and Instantaneous Rates of Change

Nurse Joy measures a patient's heart rate, in beats per minute, every thirty seconds for five minutes using a machine. The results are shown on the table below.
 

Time (in seconds)	0	30	60	90	120	150	180	210	240	270	300
Heart Rate (in BPM)	85	87	90	100	89	105	110	90	85	89	87

Using the information on the table above, find the average rate of change of the heart rate

1. Over the last two minute period.
2. Between 120 seconds and 240 seconds.

Given \(f(x)=-x^3+4\), find the slope of the secant line passing through the points on the graph of \(f\) at \(x=4.5\) and \(x=5.\)

Find the exact value of the instantaneous rate of change of \(f(x)=7+x^2\) at \(x=2.\)

Find the exact value of the instantaneous rate of change of \(f(x)=\dfrac{1}{2x+4}\) at \(x=0.\)

 An office super store sells a laptop stylus pen. The price-demand function for the stylus pen is \(p(x)=-0.2x+55\), where \(p(x)\) is the price, in dollars, per stylus pen when \(x\) stylus pens are purchased. If the cost function for the stylus pens is given by \(C(x)=15x+60\) dollars, where \(x\) is the number of stylus pens produced, find the instantaneous rate of change of profit when 225 stylus pens are produced and sold. Interpret your answer.

Find the equation of the line tangent to the graph of \(f(x)=x^2+12x-22\) at the point \((-2,-42).\)