How do you sketch the graph that satisfies f'(x)=1 when x>2, f'(x)=1 when x<2, f(2)=4?
For
For
We want
Combining the results we get:
# f(x) = { (x+a, x < 2), (4, x = 2), (x+b, x > 2) :}# where#a,b# are constantsIf we wanted a continuous solutions then we would need:
For
#x < 2 => 2+a = 4 \ \ \ \ => a=6 #
For#x > 2 => 2+b = 4 => b=2 # And so:
# f(x) = { (x6, x < 2), (4, x = 2), (x2, x > 2) :}# Which we can graph as follows:
But equally we could choose any values for the arbitrary constants
#a# and#b# , for example we could choose#a=b=0# to get
# f(x) = { (x, x < 2), (4, x = 2), (x, x > 2) :}# Which we can graph as follows:
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To sketch the graph that satisfies ( f'(x) = 1 ) when ( x > 2 ), ( f'(x) = 1 ) when ( x < 2 ), and ( f(2) = 4 ), follow these steps:

For ( x > 2 ), the function ( f'(x) = 1 ) implies that the function ( f(x) ) is increasing. So, draw a straight line with a slope of 1 for ( x > 2 ).

For ( x < 2 ), the function ( f'(x) = 1 ) implies that the function ( f(x) ) is decreasing. So, draw a straight line with a slope of 1 for ( x < 2 ).

Connect the two lines smoothly at ( x = 2 ) since ( f(2) = 4 ).

Label the point ( (2, 4) ) on the graph.
This graph represents the function ( f(x) ) that satisfies the given conditions.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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