(1 point) the point p( 0.2 , 20 ) lies on the curve y = 4 / x . let q be the point (x, 4 / x ) . a.) find the slope of the secant line pq for the following values of x. if x= 0.3, the slope of pq is: if x= 0.21, the slope of pq is: if x= 0.1, the slope of pq is: if x= 0.19, the slope of pq is: b.) based on the above results, guess the slope of the tangent line to the curve at p(0.2 , 20 ).

m = (4/x - 20)/(x - 0.2) = 4(1-5x)/x(x-0.2)

x= 0.3   | m = -66.67

x= 0.21 | m = -95.24

x= 0.1   | m = -200

x= 0.19 | m = -105.26

slope of P = -100

Step-by-step explanation:

P( 0.2 , 20 ) ∈ y = 4 / x .

Q(x, 4 / x ) .

a.) Find the slope of the secant line PQ for the following values of x.

slope: m = (y₂-y₁)/(x₂-x₁)

m = (4/x - 20)/(x - 0.2) = 4(1-5x)/x(x-0.2)

If x= 0.3, the slope of PQ is: [4(1-5.0.3)]/[0.3(0.3-0.2)] = -66.67

If x= 0.21, the slope of PQ is: [4(1-5.0.21)]/[0.21(0.21-0.2)] = -95.24

If x= 0.1, the slope of PQ is: [4(1-5.0.1)]/[0.1(0.1-0.2)] = -200

If x= 0.19, the slope of PQ is: [4(1-5.0.19)]/[0.19(0.19-0.2)] = -105.26

b.) Based on the above results, guess the slope of the tangent line to the curve at P(0.2 , 20).

Based on the results, if the slope for 0.21 is -95.24 and for 0.19 is -105.26, a difference of 10 units, as 0.2 is in the middle, it would be a difference of 10/2, so -100.

(we can confirm it using derivative and it is -100)

starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.