How do you find at equation of the tangent line to the graph #y=4+cotx-2cscx# at x=pi/2?

Answer 1

#y=-x+(pi/2+2)#

To find the equation of this tangent line we have to find the slope of this line at #x=pi/2# that is #color(red)(slope = y' at x=pi/2#
#y_(pi/2)=4+cot(pi/2)-2csc(pi/2)#
#y_(pi/2)=4+0-2*1# #y_(pi/2)=2#
this line passes through #(color(blue)(pi/2,2))#
To find the #color(red)(slope)# we should find #color(red)(y'_(pi/2))#
#y=4+cotx-2cscx#
#y'=4'+(cotx)'-2(cscx)'#
#y'=0-csc^2x-2(-cscxcotx)# #y'=-csc^2x+2cscxcotx#
#color(red)(y'_(pi/2)=-csc^2(pi/2)+2csc(pi/2)cot(pi/2)#
#color(red)(y'_(pi/2)=-1+2*0=-1)#
The Equation of the tangent line with slope#-1# and passing through #(color(blue)(pi/2,2))# : #y-color(blue)(y_0)=color(red)(y'_(pi/2))(x-color(blue)(x_0))# #y-2=-1(x-pi/2)#
#y=-x+(pi/2+2)#
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Answer 2

To find the equation of the tangent line to the graph y=4+cotx-2cscx at x=pi/2, we need to find the derivative of the function and evaluate it at x=pi/2. The derivative of y=4+cotx-2cscx is dy/dx = -csc^2(x) + csc(x)cot(x). Evaluating this at x=pi/2, we get dy/dx = -1 + 0 = -1.

The slope of the tangent line is equal to the derivative at the given point, which is -1. To find the equation of the tangent line, we use the point-slope form: y - y1 = m(x - x1), where (x1, y1) is the given point (pi/2, 4+cot(pi/2)-2csc(pi/2)) and m is the slope (-1).

Substituting the values, we have y - (4 + cot(pi/2) - 2csc(pi/2)) = -1(x - pi/2). Simplifying further, we get y - (4 + 0 - 2) = -1(x - pi/2).

Simplifying the equation, we have y - 2 = -x + pi/2. Rearranging, we get y = -x + pi/2 + 2.

Therefore, the equation of the tangent line to the graph y=4+cotx-2cscx at x=pi/2 is y = -x + (pi/2 + 2).

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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