**Math 1031 Calculus Test 2 – Answers to come**

Let f(x) = sin(2x). Only one of the following is true.

- f ” (x) = 4sin(2x)
- f ” (x) = 2cos(2x)
**f ” (x) = – 4sin(2x)**- f ” (x) = – 2cos(2x)
- A package of frozen food is removed from the freezer and t minutes after it is placed on the kitchen table, its temperature is T. Which number is bigger T’(10) or T’(100)?

**T’(10) Gets warmer faster at the beginning**- T’(100)
- they’re equal
- none of the above.

Let f(x) = x*e^{x}. Only one of the following is true.

**f’(x) = x*e**^{x}+ e^{x }- f’(x) = e
^{x} - f’(x) = x
^{2}e^{(x-1) } - none of the above.
**product rule**

- Let f(x) = 2x +2 – (5/x
^{2)}). Only one of the following is true. - f(x) concave up and increasing at x= .1
- f(x) concave up and decreasing at x= .1
**f(x) concave down and increasing at x= .1**- f(x) concave down and decreasing at x= .1
**first derivative positive, second derivative negative**

- Let f(x) = ln(x)/x. Only one of the following is true.
- f’(x) = ln(x)/x
^{2} - f’(x) = 1/x
^{2} **f’(x) = [1 – ln(x)]/x**^{2}- none of the above.
**Quotient rule**

- Joe’s salary S is a function of time, t. He says it is rising, but not as fast as it was. Which of the following is true?
- S’(t)>0, S’’(t)>0 b
**. S’(t)>0, S’’(t)<0**S’(t)<0, S’’(t)>0 d. S’(t)<0, S’’(t)<0

- Let f(x) = x
^{3 }+ 3x^{2}. Only one of the following is true. **f(x) has critical points at 0 and – 2.**- f(x) has critical points at 0 and 2
- f(x) has critical points at 0 and -1
- none of the above.
**Set derivative equal to zero, set equal to 0, and solve for x**

- Consider the function f(x) with the properties that f(2) = 19 and f’(2) = – A reasonable estimate for f(4) is which of the following.
- 21 23 c. 17 d.
**15**for each increase of 1 in x, y decreases by 2

- Let f(x) = -x
^{3 }+ 6x^{2 }+ 3. Only one of the following is true. - f’(-1) = 9.
**f’(-1) = – 15.**- f’(x) = – 9
- None of the above.
**Just plug -1 into the derivative function**

- Let f(x) = ln(2) + 7x. Only one of the following is true.
- f’(x) = ½ +7
**f’(x) = 7**- f’(x) = 1/(2 + 7)
- none of the above.
**derivative of ln(2), a constant, is 0.**

- The equation of the line tangent to y = ln(x) at x = 1,0 is
- y= x +1 y = x + e c.
**y = x – 1**d. y = x**derivative of ln(x) is 1/x, which at 1, is 1, so the line goes through 1,0 and has slope equal to 1.**

- If f(x) = (e
^{x}– 7x^{4})^{1/2}. Only one of the following is true. - then f = e
^{x/2 }– 7x^{2} **then f = u**^{1/2 }and u = e^{x}– 7x^{4}- then f = (e
^{u }– 7u^{4}) and u= x^{1/2 } - none of the above.

- What is the equation of the tangent line to f(x) = 4-x
^{2}at x = 1? - y = -2x +3 y = -x +5 c.
**y = -2x +5**d. y = -5x + 3**find derivative, -2x, plug in 1 to get a slope of -2 at 1. The line goes through 1,3.**

- Let y = ln(e
^{x}). Only one of the following is true. - f’(x) = 1/e
^{x} - f’(x) = ln(x)*e
^{x }+ e^{x}*ln(x) **f(‘x) = 1**- f’(x) = 0
**ln(e**^{x})=x, so its derivative is 1

- Let y = f(x) = 2
^{e}+ 2^{x}. Only one of the following is true. - f’(x) = e*2
^{(e-1)}+ 2^{x} **f’(x) = ln(2)*2**^{x}- f’(x) = e*2
^{(e-1)}+ ln(2)*2^{x} - 0
**2**^{e }is a constant and so has derivative 0^{ }and use the formula for derivative of a^{x}

** **

- Let y = f(x) = 1,000*2
^{(-x)}. Only one of the following is true. - f’(1,000,000) is close to 1,000
**f’(1,000,000) is close to 0**- f’(0) is close to 1,000,000
- f’(1,000) is close to 2
**2**^{-x}gets very small very fast

- Let y = f(x) = x
^{2}– 2x.^{ }The (x,y) coordinates of the relative minimum are

- 1,2

- 2,1
**1,-1**- 1,1 take derivative of f(x) to get 2x-2 and set equal to 0 to get x=1. Plug 1 into f(x).

- If f(h) = 1,000e
^{-h}– 1 gives the number of species h miles above sea level, then how fast is the number of species declining at .5 miles above sea level?

- 1000/e
^{.5}b.**– 1000/ e**c. 1000 e^{.5}^{.5}d. – 1000 e^{.5}**the derivative of e**^{-h}is –e^{-h}

- Let y = f(x) = e Only one of the following is true.
- the third derivative, f’’’(x), is equal to 3*e
^{x} **f’’’(x) = e**^{x }- f’’’(x) = e
^{x }/3 - none of the above.

- Let y = f(3) = 5 and y = g(3) = 2. Only one of the following is true.
- 2f(3) + 7g(3) = 20
- 2f(3) + 7g(3) = 28
**2f(3) – 7g(3) = – 4**- none of the above.
**just plug in**

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