4.3: The Tangent Line Approximation
- Page ID
- 106353
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This optional summary video from the textbook author might be helpful to use as a preview. Other, more detailed, supplemental videos for this section are posted at the end of the text.
Supplemental videos
Homework Exercises 4.3
WeBWorK Problems:
1. Note that you can't see the scale for the y-axis, but you can figure it out!
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Written Problems:
1. A certain function \(y=p(x)\) has its local linearization at a =3 given by \(L(x)=-2x+5\).
a. What are the values of p(3) and p′(3)? Why?
b. Estimate the value of p(2.79).
c. Suppose that p″(3)=0 and you know that p″(x)<0 for x < 3. Is your estimate in (b) too large or too small?
d. Suppose that p″(x)>0 for x > 3. Use this fact and the additional information above to sketch an accurate graph of \(y=p(x)\) near x =3. Include a sketch of \(y=L(x)\) in your work.
2. An object moving along a straight line path has a differentiable position function \(y=s(t)\); s(t) measures the object's position relative to the origin at time t. It is known that at time t =9 seconds, the object's position is s(9)=4 feet (i.e., 4 feet to the right of the origin). Furthermore, the object's instantaneous velocity at t =9 is −1.2 feet per second, and its acceleration at the same instant is 0.08 feet per second per second.
a. Use local linearity to estimate the position of the object at t =9.34.
b. Is your estimate likely too large or too small? Why?
c. In everyday language, describe the behavior of the moving object at t=9. Is it moving toward the origin or away from it? Is its velocity increasing or decreasing? Include a sketch of the graph of the position function in your explanation.