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

Math 1031 Calculus Test 2 – Answers to come

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

1. f ” (x) = 4sin(2x)
2. f ” (x) = 2cos(2x)
3. f ” (x) = – 4sin(2x)
4. f ” (x) = – 2cos(2x)
7. 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)?

2. T’(10) Gets warmer faster at the beginning
3. T’(100)
4. they’re equal
5. none of the above.

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

3. f’(x) = x*e^x + e^x  
4. f’(x) = e^x
5. f’(x) = x²e^(x-1)
6. none of the above. product rule

4. Let f(x) = 2x +2 – (5/x²⁾). Only one of the following is true.
5. f(x) concave up and increasing at x= .1
6. f(x) concave up and decreasing at x= .1
7. f(x) concave down and increasing at x= .1
8. f(x) concave down and decreasing at x= .1 first derivative positive, second derivative negative

5. Let f(x) = ln(x)/x. Only one of the following is true.
6. f’(x) = ln(x)/x²
7. f’(x) = 1/x²
8. f’(x) = [1 – ln(x)]/x²
9. none of the above. Quotient rule

6. 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?
7. 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

7. Let f(x) = x³ + 3x². Only one of the following is true.
8. f(x) has critical points at 0 and – 2.
9. f(x) has critical points at 0 and 2
10. f(x) has critical points at 0 and -1
11. none of the above. Set derivative equal to zero, set equal to 0, and solve for x

8. 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.
9. 21 23    c. 17     d. 15            for each increase of 1 in x, y decreases by 2

9. Let f(x) = -x³ + 6x² + 3. Only one of the following is true.
10. f’(-1) = 9.
11. f’(-1) = – 15.
12. f’(x) = – 9
13. None of the above. Just plug -1 into the derivative function

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

11. The equation of the line tangent to y = ln(x) at x = 1,0 is
12. 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.

12. If f(x) = (e^x – 7x⁴)^1/2. Only one of the following is true.
13. then f = e^x/2 – 7x²
14. then f = u^1/2 and u = e^x – 7x⁴
15. then f = (e^u – 7u⁴) and u= x^1/2
16. none of the above.

13. What is the equation of the tangent line to f(x) = 4-x² at x = 1?
14. 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.

14. Let y = ln(e^x). Only one of the following is true.
15. f’(x) = 1/e^x
16. f’(x) = ln(x)*e^x + e^x*ln(x)
17. f(‘x) = 1
18. f’(x) = 0 ln(e^x)=x, so its derivative is 1

15. Let y = f(x) = 2^e + 2^x. Only one of the following is true.
16. f’(x) = e*2^(e-1) + 2^x
17. f’(x) = ln(2)*2^x
18. f’(x) = e*2^(e-1) + ln(2)*2^x
19. 0 2^e is a constant and so has derivative 0  and use the formula for derivative of a^x 

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

1. Let y = f(x) = x² – 2x.  The (x,y) coordinates of the relative minimum are

1. 1,2

1. 2,1
2. 1,-1
3. 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).

18. 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?

1. 1000/e^.5    b. – 1000/ e^.5          c.  1000 e^.5    d. – 1000 e^.5    the derivative of e^-h is –e^-h

19. Let y = f(x) = e Only one of the following is true.
20. the third derivative, f’’’(x), is equal to 3*e^x
21. f’’’(x) = e^x
22. f’’’(x) = e^x /3
23. none of the above.

20. Let y = f(3) = 5 and y = g(3) = 2. Only one of the following is true.
21. 2f(3) + 7g(3) = 20
22. 2f(3) + 7g(3) = 28
23. 2f(3) – 7g(3) = – 4
24. none of the above. just plug in