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“Graphs” is an easy and scoring section in the exams, and something that you can use a good deal in your future PowerPoint Presentations:
Question 1: If f and g are the functions whose graphs are shown, let ux=f(x)g(x) and v(x) = f(x)/g(x)
Explanation: To find the derivative of the function u(x) and v(x), we can use the product rule and the quotient rule of the derivative respectively.
So, if ux=f(x)g(x).
Then, we know that u'x=f'xgx+g'xf(x)
After analyzing the graph, we can see that: f(1)=3, f'(1)=3. And, g(1)=2, g'(1) = -1.
So, u'(1)=f'(1)g(1)+g'(1)f(1)
Let us plug the values.
u'1=3×2+-1×3=6-3 = 3
So, u'1=3
We have given v(x) = f(x)/g(x).
Then, using the quotient rule,
v'x=gxf'x-fxg'x/(g(x))^2
Looking at the graph, we have: f(5)=2, f'(5)=-1/4. And, g(5)=2, g'(5)=1.
Now, let us plug these values.
v'5=g(5)f'(5)-f(5)g'(5)/g(5)^2
v'5=2×(-1/4)-2×1/(1)^2= (-1/2)-2/1=-5/2
Hence, v'5=-5/2
Question 2: Which graphs represent an object moving in the positive direction?
An object moves in the positive direction if its velocity is positive. The direction of motion cannot be stated by analyzing the slope of the velocity-time graph.
Example:
Question 3: Which graphs represent an object moving in the negative direction?
The direction of an object would be negative if it has a negative velocity. The direction of motion cannot be defined by the slope of the velocity-time graph.
The positive or negative slope of the velocity-time graph does not state the positive or negative direction. It just describes how velocity is changing with respect to time.
Example:
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