1)
Find the slope of the tangent line to 𝑦 = 𝑥 3 at 𝑥 = 2. Then find and compare this to the
average rate of change of 𝑦 = 𝑥 3 on the interval [1,3]
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5
2) Find the instantaneous rate of change of 𝑓(𝑥) = 𝑥−3 𝑎𝑡 𝑥 = 2. Then write the equation of the
tangent line in both Point Slope Form, and Slope Intercept Form. Then use a graphing calculator
to sketch both.
3) A particle travels along the x-axis so that its position (P) is given by 𝑃 = 𝑡 2 − 6𝑡 + 2. Find the
speed of the particle at 𝑡 = 4. Then find the speed of the particle at 𝑡 = 1. Speculate on the
meaning of this second answer.
4) Write the equation of the normal line to 𝑦 = √𝑥 + 4 at 𝑥 = 0
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5) Find the slop of the tangent line to ℎ(𝑥) = 𝑥 + 𝑥 at 𝑥 = 1
4𝑒 2(𝑥+ℎ) −4𝑒 2𝑥
.
ℎ
ℎ→0
6) The following is finding the slope of a tangent line of some function: lim
What is
the function? Use a calculator to find this slope by choosing a very small value for ℎ, and
evaluating it at 𝑥 = −1
7) Multiple Choice: Which of the following could be used to find the slope of the tangent line to
𝜋
𝑔(𝑥) = sin(𝑥) at 𝑥 = 6
a) lim
ℎ→0
sin(𝑥)+ℎ−
ℎ
1
2
b) lim
ℎ→0
𝜋
6
sin( +ℎ)−
ℎ
1
2
c) lim
ℎ→0
1
2
sin(ℎ)−
ℎ
d) lim
ℎ→0
𝜋
6
𝜋
−0
6
sin( )−sin(0)
8) If someone measures your pulse (presumably in beats per minute) is this an average rate of
change, or an instantaneous rate of change?
9) If you are driving and you look at your dashboard to determine your speed, is this an average
rate of change, or an instantaneous rate of change?
10) Identify the problem with the following question: “What is the average rate of change of
𝑓(𝑥) = 𝑥 2 at 𝑥 = 10 ?”
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11) Use the following picture to view 𝑦 = 3 𝑥 2 . Print this page dark enough that you can still see
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the grid lines. Use a straight edge to sketch a tangent line to the graph of 𝑦 = 3 𝑥 2 at the point
where 𝑥 = 3. To the best of your ability, find two points on this line, so that you can calculate
𝑦 −𝑦
its slope the way you did in Algebra I (𝑥2 −𝑥1 ).
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1
Then use the techniques we have in class to find the slope of the tangent line and see how
closely the two will agree.