# Thread: 5 CONCEPTUAL problems I need help with, thank you.

1. ## 5 CONCEPTUAL problems I need help with, thank you.

These are not difficult subjects, but they are tricky problems. To prepare for a test our professor gave us some practice problems to make sure we understand the concepts. I got most of them, but here are a few I can't figure out.

45) Consider the curve defined by -8x2 +5xy+y3 =-149. Write and equation for the line tangent to f at the point (4, -1) and using this line approximate f(4.2).

• a) -0.4
• b) 0.4
• c) -0.373
• d) 0.373
• e) -0.6

CONCEPT: linearization, (and implicit differentiation).

39) If f(x) = x3 +x and g(x) is the inverse function of f(x), then g'(1) =

• a) -0.5
• b) 0.003
• c) 0.077
• d) 0.25
• e) 0.417
• THIS ONE I FIGURED OUT ON WOLFRAM ALPHA, BUT CANNOT DO ON MY TI-89 CALCULATOR

37) An object traveling in a straight line has position s(t) at time t. If the initial position is s(0) = 3 and the velocity of the object is v(t) = (1+2t2 )(1/3) , what is the position of the object at time t=2?

• a)7.312
• b)5.933
• c)2.933
• d)8.312
• e)24
• THE VELOCITY FUNCTION IS EXACTLY HOW THEY HAVE IT. WITHOUT AN EXTRA t IT IS HARD TO INTEGRATE, BUT I THINK I'M SUPPOSE TO BE ABLE TO FIGURE IT OUT CONCEPTUALLY.

18) Let g be a twice-differentiable function with g'(x) > 0 and g''(x) > 0 for all real numbers x, such that g(4) = 12 and g(5) = 18. Of the following, which is a possible value for g(6)?

• a)15
• b)18
• c)21
• d)24
• e)27

16) The function f is differentiable and has values as shown in the table below. Both f and f' are strictly increasing on the interval 0<=x<=5. Which of the following could be the value of f'(3)?

TABLE:

x = 2.5 | 2.8 | 3.0| 3.1

f(x)= 31.25 | 39.2 | 45| 48.05

• a)20
• b)27.5
• c)29
• d)30
• e)48.05

THANK YOU THANK YOU THANK YOU, FOR ANY HELP!!!

2. ## Re: 5 CONCEPTUAL problems I need help with, thank you.

Originally Posted by jimmeyers
These are not difficult subjects, but they are tricky problems. To prepare for a test our professor gave us some practice problems to make sure we understand the concepts. I got most of them, but here are a few I can't figure out.

45) Consider the curve defined by -8x^2 +5xy+y^3 =-149. Write and equation for the line tangent to f at the point (4, -1) and using this line approximate f(4.2).

a straight-forward implicit derivative ... what did you get for dy/dx ?

• a) -0.4
• b) 0.4
• c) -0.373
• d) 0.373
• e) -0.6

CONCEPT: linearization, (and implicit differentiation).

39) If f(x) = x3 +x and g(x) is the inverse function of f(x), then g'(1) =

• a) -0.5
• b) 0.003
• c) 0.077
• d) 0.25
• e) 0.417
• THIS ONE I FIGURED OUT ON WOLFRAM ALPHA, BUT CANNOT DO ON MY TI-89 CALCULATOR

if f and g are inverses, then f[g(x)] = x ... take the derivative of this equation w/r to x and determine a relationship between the derivatives of the inverse functions

37) An object traveling in a straight line has position s(t) at time t. If the initial position is s(0) = 3 and the velocity of the object is v(t) = (1+2t2 )(1/3) , what is the position of the object at time t=2?

fundamental theorem ... $\displaystyle s(t_2) - s(t_1) = \int_{t_1}^{t_2} v(t) \, dt$

• a)7.312
• b)5.933
• c)2.933
• d)8.312
• e)24
• THE VELOCITY FUNCTION IS EXACTLY HOW THEY HAVE IT. WITHOUT AN EXTRA t IT IS HARD TO INTEGRATE, BUT I THINK I'M SUPPOSE TO BE ABLE TO FIGURE IT OUT CONCEPTUALLY.

18) Let g be a twice-differentiable function with g'(x) > 0 and g''(x) > 0 for all real numbers x, such that g(4) = 12 and g(5) = 18. Of the following, which is a possible value for g(6)?

g'(x) > 0 and g''(x) > 0 ... what does this tell you about the rate of change of g ?

• a)15
• b)18
• c)21
• d)24
• e)27

16) The function f is differentiable and has values as shown in the table below. Both f and f' are strictly increasing on the interval 0<=x<=5. Which of the following could be the value of f'(3)?

same concept as the previous question

TABLE:

x = 2.5 | 2.8 | 3.0| 3.1

f(x)= 31.25 | 39.2 | 45| 48.05

• a)20
• b)27.5
• c)29
• d)30
• e)48.05
...

3. ## Re: 5 CONCEPTUAL problems I need help with, thank you.

Thank you for the response!

For #39 I am still a little lost. First, I wrote it incorrectly: f(x) should be (x^3 + x). But also, I understand that g'(x) = 1/f'(g(x)) and this would suggest it goes the other way, f'(x) = 1/g'(f(x)). If I take the derivative of f(x) I get 3x^2+1 set that equal to 1/g'(f(x)) or 1/g'(1) then flip, I get g'(1) = 1/(3x^2+1). If I put 1 in for x I get 1/4 or .25, which is one of the options but incorrect.

For #45, when I use the ImpDif function on my calculator (TI-89) it gives me 16x/3y^2. When I did it by hand I got the correct derivative and it all worked out (A). I have tried it a couple times - do you have any experience with these calculators? (Or perhaps ImpDif is the wrong command.)

For #37, thank you, I had only tried it as an indefinite - it didn't cross my mind to try it as a definite integral which makes me think I am missing some basic intuition with derivatives/integrals.

For 16 and 18, I figured them out using what I know about the first and second derivatives, but I felt like that was more of a good guess. Especially with 16. I believe, and correct me if I'm wrong, that I can only find the secant lines but those should still be getting steeper - does this sound right? (That is, unless I take the limit and algebraically work out the actual definition of a derivative - is this what you would do?)

Anyway, I really appreciate the help! Thank you!

4. ## Re: 5 CONCEPTUAL problems I need help with, thank you.

Originally Posted by jimmeyers
For #39 I am still a little lost. First, I wrote it incorrectly: f(x) should be (x^3 + x). But also, I understand that g'(x) = 1/f'(g(x)) and this would suggest it goes the other way, f'(x) = 1/g'(f(x)). If I take the derivative of f(x) I get 3x^2+1 set that equal to 1/g'(f(x)) or 1/g'(1) then flip, I get g'(1) = 1/(3x^2+1). If I put 1 in for x I get 1/4 or .25, which is one of the options but incorrect.
$\displaystyle g'(1) = \frac{1}{f'[g(1)]}$

since f and g are inverses, $\displaystyle f(a) = 1$ , $\displaystyle g(1) = a$

$\displaystyle x^3+x = 1$ at $\displaystyle x \approx 0.682$ so $\displaystyle a = 0.682$

$\displaystyle g'(1) = \frac{1}{f'(a)} = \frac{1}{3a^2+1} = 0.417$

For #45, when I use the ImpDif function on my calculator (TI-89) it gives me 16x/3y^2. When I did it by hand I got the correct derivative and it all worked out (A). I have tried it a couple times - do you have any experience with these calculators? (Or perhaps ImpDif is the wrong command.)
I did this by hand (as you should) ...

$\displaystyle \frac{dy}{dx} = \frac{16x-5y}{5x+3y^2}$

link w/ instructions for doing it on an 89 ...

Module 13 - Implicit Differentiation

For 16 and 18, I figured them out using what I know about the first and second derivatives, but I felt like that was more of a good guess. Especially with 16. I believe, and correct me if I'm wrong, that I can only find the secant lines but those should still be getting steeper - does this sound right? (That is, unless I take the limit and algebraically work out the actual definition of a derivative - is this what you would do?)
correct