2.1E: Exercises
- Page ID
- 116546
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For exercises 1 - 3 , points \(P(1,2)\) and \(Q(x,y)\) are on the graph of the function \(f(x)=x^2+1\).
- [Technology Required] Complete the following table with the appropriate values: \(y\)-coordinate of \(Q\), the point \(Q(x,y)\), and the slope of the secant line passing through points \(P\) and \(Q\). Round your answer to eight significant digits.
| \(x\) | \(y\) | \(Q(x,y)\) | \(m_{sec}\) |
|---|---|---|---|
| 1.1 | |||
| 1.01 | |||
| 1.001 | |||
| 1.0001 |
- Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to \(f\) at \(x=1\).
- Use the value in the preceding exercise to find the equation of the tangent line at point \(P\). Graph \(f(x)\) and the tangent line.
- Answer
- \(y=2x\)
For the exercises 4-6, points \(P(1,1)\) and \(Q(x,y)\) are on the graph of the function \(f(x)=x^3\).
- [Technology Required] Complete the following table with the appropriate values: \(y\)-coordinate of \(Q\), the point \(Q(x,y)\), and the slope of the secant line passing through points \(P\) and \(Q\). Round your answer to eight significant digits.
| \(x\) | \(y\) | \(Q(x,y)\) | \(m_{sec}\) |
|---|---|---|---|
| 1.1 | |||
| 1.01 | |||
| 1.001 | |||
| 1.0001 |
- Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to \(f\) at \(x=1\).
- Answer
- \(3\)
- Use the value in the preceding exercise to find the equation of the tangent line at point \(P\). Graph \(f(x)\) and the tangent line.
For the exercises 7 - 9, points \(P(4,2)\) and \(Q(x,y)\) are on the graph of the function \(f(x)=\sqrt{x}\).
7) [Technology Required] Complete the following table with the appropriate values: \(y\)-coordinate of \(Q\), the point \(Q(x,y)\), and the slope of the secant line passing through points \(P\) and \(Q\). Round your answer to eight significant digits.
| \(x\) | \(y\) | \(Q(x,y)\) | \(m_{sec}\) |
|---|---|---|---|
| 4.1 | a. | e. | i. |
| 4.01 | b. | f. | j. |
| 4.001 | c. | g. | k. |
| 4.0001 | d. | h. | l. |
- Answer
- a. 2.0248457
b. 2.0024984
c. 2.0002500
d. 2.0000250
e. (4.1000000,2.0248457)
f. (4.0100000,2.0024984)
g. (4.0010000,2.0002500)
h. (4.00010000,2.0000250)
i. 0.24845673
j. 0.24984395
k. 0.24998438
l. 0.24999844
8) Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to \(f\) at \(x=4\).
9) Use the value in the preceding exercise to find the equation of the tangent line at point \(P\).
- Answer
- \(y=\frac{x}{4}+1\)
For exercises 10 - 12, points \(P(1.5,0)\) and \(Q(ϕ,y)\) are on the graph of the function \(f(ϕ)=\cos( \pi ϕ)\).
10) [Technology Required] Complete the following table with the appropriate values: \(y\)-coordinate of \(Q\), the point \(Q(ϕ,y)\), and the slope of the secant line passing through points \(P\) and \(Q\). Round your answer to eight significant digits.
| \(x\) | \(y\) | \(Q(ϕ,y)\) | \(m_{sec}\) |
|---|---|---|---|
| 1.4 | a. | e. | i. |
| 1.49 | b. | f. | j. |
| 1.499 | c. | g. | k. |
| 1.4999 | d. | h. | l. |
11) Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at \(ϕ=1.5\).
- Answer
- \( \pi \)
12) Use the value in the preceding exercise to find the equation of the tangent line at point \(P\).
For exercises 13 - 15, points \(P(−1,−1)\) and \(Q(x,y)\) are on the graph of the function \(f(x)=\frac{1}{x}\).
13) [Technology Required] Complete the following table with the appropriate values: \(y\)-coordinate of \(Q\), the point \(Q(x,y)\), and the slope of the secant line passing through points \(P\) and \(Q\). Round your answer to eight significant digits.
| \(x\) | \(y\) | \(Q(x,y)\) | \(m_{sec}\) |
|---|---|---|---|
| -1.05 | a. | e. | i. |
| -1.01 | b. | f. | j. |
| -1.005 | c. | g. | k. |
| -1.001 | d. | h. | l. |
- Answer
- a. −0.95238095
b. −0.99009901
c. −0.99502488
d. −0.99900100
e. (−1;.0500000,−0;.95238095)
f. (−1;.0100000,−0;.9909901)
g. (−1;.0050000,−0;.99502488)
h. (1.0010000,−0;.99900100)
i. −0.95238095
j. −0.99009901
k. −0.99502488
l. −0.99900100
14) Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to \(f\) at \(x=−1\).
15) Use the value in the preceding exercise to find the equation of the tangent line at point \(P\).
- Answer
- \(y=−x−2\)
For exercises 16 - 17, the position function of a ball dropped from the top of a 200-meter tall building is given by \(s(t)=200−4.9t^2\), where position \(s\) is measured in meters and time \(t\) is measured in seconds. Round your answer to eight significant digits.
16) [Technology Required] Compute the average velocity of the ball over the given time intervals.
a. [4.99,5]
b. [5,5.01]
c. [4.999,5]
d. [5,5.001]
17) Use the preceding exercise to guess the instantaneous velocity of the ball at \(t=5\) sec.
- Answer
- \(−49\) m/sec (velocity of the ball is 49 m/sec downward)
For exercises 18 - 19, consider a stone tossed into the air from ground level with an initial velocity of 15 m/sec. Its height in meters at time t seconds is \(h(t)=15t−4.9t^2\).
18) [Technology Required] Compute the average velocity of the stone over the given time intervals.
a. [1,1.05]
b. [1,1.01]
c. [1,1.005]
d. [1,1.001]
19) Use the preceding exercise to guess the instantaneous velocity of the stone at \(t=1\) sec.
- Answer
- \(5.2\) m/sec
For exercises 20 - 21, consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by \(h(t)=600+78.4t−4.9t^2\), where \(t\) is measured in seconds.
20) [Technology Required] Compute the average velocity of the rocket over the given time intervals.
a. [9,9.01]
b. [8.99,9]
c. [9,9.001]
d. [8.999,9]
21) Use the preceding exercise to guess the instantaneous velocity of the rocket at \(t=9\) sec.
- Answer
- \(-9.8\) m/sec
For exercises, consider an athlete running a 40-m dash. The position of the athlete is given by \(d(t)=\frac{t^3}{6}+4t\), where \(d\) is the position in meters and \(t\) is the time elapsed, measured in seconds.
22) [Technology Required] Compute the average velocity of the runner over the given time intervals.
a. [1.95,2.05]
b. [1.995,2.005]
c. [1.9995,2.0005]
d. [2,2.00001]
23) Use the preceding exercise to guess the instantaneous velocity of the runner at \(t=2\) sec.
- Answer
- \(6\) m/sec
24) Estimate the slope of the tangent line to the function at \( t = 1 \) for the given function.
a. \( \sinh{(x)} \)
b. \( \cosh{(x)} \)
c. \( \tanh{(x)} \)