Please check my answers (and give correct ones) I tend to make dumb mistakes so I'm trying to work on that (sorry in advance) and my teacher never gives us the answer sheet so I have a hard time knowing what to study TT

Question

1. The velocity of a runner is given below:
(gyazo.com/b9469e974161583e05f5ceb09854e931)
Using trapezoids, estimate the total distance the runner travels from t=0 to t=6 seconds.
40
26
24 <--- !
17

2. If f(x) = e^(2ln(x^3 + 5)), then f’(x)
e^(2ln(x^3 + 5)) <--- !
(2)/(x^3 + 5) e^(2ln(x^3 + 5))
6ln(x^2) e^(2ln(x^3 + 5))
6x^2 (x^3 + 5)

3. ∫ (pi on top, pi/2 on bottom) (cos theta)/(sqrt(1+sin theta)) d theta
-2(sqrt(2) -1) <--- !
-2 sqrt(2)
2(sqrt(2) - 1)
2(sqrt(2) + 1)

4. Given that f(-0.5) = 2 f’(-0.5) = 4, using a tangent line approximation you would estimate f(0) to be:
1
-2
-3
4 <--- !

5. If all the critical values of f(x) are shown in the chart below, which of the following could be values for f(x) at x = 1(1/2), 2(1/2), and 3(1/2)?
Chart: (gyazo.com/214d78ec91b2d8c3fea6f96806915a18)

Answer choices: (gyazo.com/421b6529a3456171a0e35d33c71e8d0d)

Second choice <--- !

6. Which of the following will not give the instantaneous rate of change of f(x) with respect to x when x = 3?
(f(3)-f(3-0.1))/(0.1)
Lim as h->0 (f(3+h)-f(3))/(h)
Lim as h->0 (f(3+h)-f(3-h))/(2h) <--- !
Lim as h->3 (f(3)-f(x))/(3-x)

7. Evaluate lim x->inf (20lnx)/(x)
0 <--- !
1
20
DNE

8. If y=2 f(g(x)), then (d^2 y)/(dx^2)=
f”(g(x)) * g’(x)
f”(g(x)) * [g’(x)]^2 <--- !
2f”(g(x)) * [g’(x)]^2
2f”(g(x)) * [g’(x)]^2 + 2f’(g(x)) * g”(x)

9. A particle moves along a line so that at time t seconds, where 0 < t < pi, its position is given in meters by,
s(t) = -4sint - t/2 + 10
What is the acceleration of the particle the first time its velocity equals zero?
-5.197 (m)/(sec^2)
1.323 (m)/(sec^2)
2.550 (m)/(sec^2) <--- !
3.969 (m)/(sec^2)

10. The rectangle shown below has vertices of (0,0), (pi/2, 0), (pi/2,1), and (0,1):
(gyazo.com/fc46f91ee008c73a50f1c4fbd32b865b)
What percent of the rectangle is shaded?
36.3% <--- !
63.7%
43.2%
31.4%

11. If the graph of y=(ax-b)/(x-c) has horizontal asymptote y=5 and a vertical asymptote x=-2, then b cannot be equal to:
12
-5 <--- !
-10
0

12. If dy/dx = (1+x)/(xy), x>0, and y=-4 when x=1, when x=3, y=
4.711 and -4.711
3.898 and -3.898
-4.711 only <--- !
3.898 only

13. If the function f’ has a continuous derivative on [0,c], then ∫ (c on top, 0 on bottom) f”(x) dx =
f”(c) - f”(0) <--- !
|f(c) - f(0)|
f’(c)
f’(c) - f’(0)

14. Use a left-endpoint Riemann sum approximation with four subintervals to evaluate ∫ (8 on top, 0 on bottom) g(x)dx
gyazo.com/7966c6ce1230cda3641cb3de1645a416
11
-5.5
-11 <--- !
-13.5

15. If q is positive and increasing, for what value of q is the rate of increase of q^3 twelve times that of the rate increase of 6?
2
3
12 <--- !
36

16. Given that g is the inverse function of f, and f(3) = 4, and f’(3) = 5, then g’(4) =
-1/5
-3/5
3/5 <--- !
1/5

17. What is the minimum value of f(x) = xe^x
-1/e
-e
-1 <--- !
0

18. Which of these curves is decreasing at an increase rate?
(1 and 2) (gyazo.com/636fce52943f3102ddbfeae086ce4763)
(2 and 3) (gyazo.com/227604cdbe1bada60e4047a5a4939eb9)

Graph 2 <--- !

19. If f(x) is continuous for all x, which of the following integrals necessarily have the same value?
(gyazo.com/7e8fba1d66d58a87ef3f8757f98dc7c6)
I and II only
I and III only <--- !
I, II and IV only
II, III, and IV only

20. The acceleration (m/sec^2) of an object along a straight line is given in the following graph:
(gyazo.com/b79242a86250a6c6854a7681098aaa37)
Assuming that the object has an initial velocity of 5 m/sec, how far does the particle move during the time 0<= t<= 3?
27.0 m <--- !
28.5 m
33 m
53.5 m

21. f(x) is continuous for -0.5 <= x <= -0.2 and also has the following values:
(gyazo.com/68f74b80c3b9035e6b0176a97fec8e2a)
Which of the following may be true about f(x)?
(gyazo.com/506fab0be98347a8251e074f497b7d0b)
II only
III only <--- !
I and III only
I,II, and III

22. Find the equation of a line tangent to the curve xy = sqrt(xy-1) +1 at the point (1,2)
y = -2x + 4 <--- !
y = 2.667 - 1.667x
y = 2.667 - 1.333x
y = 2-x

Answer 1

#1 I get 40. How did you get 24?
#2 recall that e^(lnx) = x
So, you have f(x) = (x^3+5)^2
Now what is your answer?
You have just said that d/dx e^u = e^u, but you forgot the chain rule.
#3 That's not what I get. Let u = 1+sinθ and you have
∫ du/√u
#4 ok
#5 ?? Critical values are where f' = 0. The able shows all nonzero values for f'.
#6 the limits are all instantaneous values...
#7 ok
#8 don't forget to use the product rule on y'
dy/dx = 2f'(g)*g'
#9 Hmmm. I get v=0 at t = 97.18° so ...
#10 ok
#11 You have (5x-b)/(x+2)
since there is a vertical asymptote, the top and bottom cannot both be zero
#12 ok
#13 nope. ∫f" = f'
#14 ok
#15 I get 2. How did you get 12?
#16 nope 1/5 - how did you get 3/5?
#17 no. you want f(-1)
#18 ok
#19 ok
#20 I get 33 - how did you get 27?
start with a(t) = 1+3t
#21 I got I,II,III what is your reasoning?
#22 ok