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You'd need to:
-Differentiate w(x) with respect to x, to obtain an expression for w'(x)
-Find the values of x for which w'(x) = 0
-Substitute these values of x into your equation for w''(x) to determine which values of x correspond to a minimum and which correspond to a maximum
Correct.
One other approach you could have used is to calculate w'(x) to determine its stationary points, then evaluating w''(x) at these points to determine which of these is a maximum and which of these is a minimum. However, the approach taken above is a lot quicker and perhaps more intuitive.
(-3/a) + (4/b) = 1
I say a = d and b = d.
What happens when you substitute a = d and b = d into the equation above?
For part (a) you're asked to determine the acceleration in terms of t, so you'll also need to calculate s''(t).
Other parts are correct.
You want to find how many times you need to differentiate y until you end up with your original function y again.
Yes, you need to find the second derivative and then add that to y.
Correct.
Correct.
Yes -- finally, what are the maximum and minimum values of w(x)?
You might like to use the fact that:
sec(t)*cos(t) = 1
and
sec(t)*sin(t) = tan(t)
Can you see why these are true?
Try plotting the point (3, -2) on a graph -- then draw a horizontal line through it. Once you've done this, can you identify some other points on the line?
You have
(-3/a) + (4/b) = 1
and
I say a = d and b = d.
How can you eliminate a and b to determine the (numerical) value of d?
Taking the derivative in terms of t, I get
V = -(1/8) sin(t).
I don't know how to find the maximum speed from this point on.
What is the range of possible values of sin(t)?
What is the range of possible values of -(1/8)sin(t)?
What is the range of possible values of V?
For part (a), I must evaluate R(t) at t = 1 and t = 2. Yes?
No. You have been given four dates, not two -- so what are the values of t at these dates?
For part (b), I get R'(t) = cos (t) + 0.3.
Correct.
For part (c), what particular value of t are you taking about?
The value of t that is mentioned in the question.
What are the smallest and largest values of 2 + cos(x)?
For part (a), what is the value of t for the range of dates you've given?
For part (b), you want to calculate the rate of change of R(t) with respect to t. In other words, you'll need to determine R'(t).
Part (c) just involves calculating R'(t) at that particular value of t.
You have been given the displacement, s, of the object in terms of time, t. If you differentiate this with respect to t, you'll get the object's velocity in terms of time. How would you find the maximum speed from here?
Note that this equation describes a phenomenon known as simple harmonic motion -- if you've come across this before (and know the sort of motion it's describing), you can have a think about how the object's speed varies between its points of maximum and minimum displacement, which is a shortcut to the answer.
To find the minimum and maximum value of w, there are a few different approaches you could take.
The question is asking: what are the smallest and largest possible values of ?Here's a starting point: what are the smallest and largest possible values of ?2. The slope of the line y = mx + b is the coefficient of x or m.
Correct -- although it's better to say that it's the coefficient of x, as m doesn't usually represent a variable in the equation of a straight line.
1. The slopes are negative reciprocal of each other.
No, that's only true for perpendicular lines.
Bob has helpfully provided a picture to illustrate what is going on -- have you read and understood his explanation?
I say a = d and b = d.
Correct.
So what's the value of d?
Correct.
So you want a horizontal line which passes through (3, -2) and a vertical line which passes through (2, 14). What other points might lie on these lines? Is there a pattern?
1. I think a and b are constants. In that case, a = -3 and b = 4.
Correct.
2. You are saying to substitute a = -3 and b = 4 into
(x/a) + (y/b) = 1
(-3/a) + (4/b) = 1
Is this what you mean?
Yes.
Also, where does the x-intercept: (d, 0), d ≠ 0 and the
y-intercept: (0, d), d ≠ 0 come into play?
What does this tell you about the values of a and b?
Yes is the answer to your first question.
For your second, have a think about the shape of your graph. What does it look like? And what do you think might represent the 'arch support' here?
For #1, what are a and b in this case?
For #2, you're given a value of x and a value of y. What happens when you substitute those into the equation? Moreover, what are the values of a and b?
Consider a line with m = 0 (i.e. zero gradient). What does that look like?
Now consider a line with undefined gradient*. What does that look like?
*Note that the term 'undefined gradient' can have a different interpretation but since we know it is a straight line we can get away with it here.