Math 2210 - 01 Calculus I Midterm 1 Zadock Reid¶
Last name: Wibowo First name: James Score:
Instructions: Please answer each question carefully. Make sure to write legibly and always circle your answer. Remember: the correct answer is ONLY worth 1 point, and the correct work is worth up to 39 points.¶
1. a) Find a linear function that generates the values in the table.¶
| x | 5.2 | 5.3 | 5.4 | 5.5 | 5.6 |
|---|---|---|---|---|---|
| y | 27.8 | 29.2 | 30.6 | 32.0 | 33.4 |
\[m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$$
$$\frac{y_2-y_1}{x_2-x_1}\Rightarrow\frac{33.4-27.8}{5.6-5.2}=14$$
$$y=14x\Rightarrow y=14(5.2)=72.8\Rightarrow27.8-72.8=-45$$
$$\boxed{y=14x-45}\]
b) A 325 mg aspirin has a half-life of \(H\) hours in a patient's body.¶
i) How long does it take for the quantity of aspirin in the patient's body to be reduced to 162.5 mg? To 81.25 mg? To 40.625 mg? (Note that \(162.5 = 325/2\), etc. Your answers will involve \(H\).)¶
\[\boxed{\textrm{Quantity (mg)}=325/2^H}\]
ii) How many times does the quantity of aspirin, A mg, in the body halve in t hours? Use this to give a formula for A after t hours.¶
The quantity halves every hour \(\(\boxed{A=A/2^t}\)\)
2. a) By graphing, estimate \(\lim_{x\rightarrow\infty}(1+\frac{1}{x})^x\). You should recognize the answer you get.¶
\(\(\boxed{y=e}\)\)
b) Investigate \(\lim_{x\rightarrow\infty}(1+\frac{1}{x})^x\) numerically.¶
| x | 1 | 5 | 10 | 100 | 1000 | 10000 |
|---|---|---|---|---|---|---|
| y | 2 | 2.48832 | 2.593724601... | 2.70481382942... | 2.71692393224... | 2.71814592683... |
3. a) \(\(\lim_{h\rightarrow0}\frac{\sqrt{1+h}-1}{h}\textrm{, let }t=\sqrt{1+h}\)\)¶
\[\lim_{h\rightarrow0}\frac{\sqrt{1+h}-1}{h}\textrm{, }t=\sqrt{1+h}\Rightarrow \lim_{h\rightarrow0}\frac{t-1}{h}\textrm{, }t^2=1+h\Rightarrow\lim_{t^2-1\rightarrow0}\frac{t-1}{t^2-1}\textrm{, }t^2=1+h\Downarrow$$
$$\lim_{t^2\rightarrow1}\frac{t-1}{t^2-1}\Rightarrow\lim_{t^2\rightarrow1}\frac{t-1}{(t+1)(t-1))}\Rightarrow\lim_{t^2\rightarrow1}\frac{1}{t+1}\Rightarrow\frac{1}{1+1}\Rightarrow\boxed{\frac{1}{2}}\]
b) Explain what is wrong with the statement.\(\(\textrm{if }f(x)=\frac{x^2-1}{x+1}\textrm{ and }g(x)=x-1\textrm{, then }f=g\)\)¶
\[\boxed{\textrm{False, because }\frac{x^2-1}{x+1}\textrm{ where }x=-1\textrm{ will be undefined and }g(x)=x-1\textrm{ where }x=-1\textrm{ will be }-2}\]
1. A ball is tossed into the air from a bridge, and its height, y (in feet), above thev
ground t seconds after it is thrown is given by
𝑦=𝑓(𝑡)=− 16 𝑡^2 + 50 𝑡+ 36
a) How high above the ground is the bridge?
b) What is the average velocity of the ball for the first second?
c) Approximate the velocity of the ball at t = 1 second.
d) Graph f, and determine the maximum height the ball reaches. What is the velocity at the time the ball is at the peak?
e) Use the graph to decide at what time, t, the ball reaches its maximum height.
- Find the derivative of the function 𝑓(𝑥)=^1 𝑥 , x ≠ 0, by the definition of the derivative of
a function.