For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t x=et ,y=(t-1)2 ,at(1,1)

y+2x = 3

Step-by-step explanation:

The formula for finding the equation of a tangent line is expressed using the point slope equation y-y0 = m(x-x0) where:

m is the slope of the line and (x0, y0) is a point on the line.

m = dy/dx = dy/dt ÷ dx/dt

dy/dt = 2(t-1)

dx/dt = e^t

m = 2(t-1)/e^t

If x = e^t, and x = 1

1 = e^t

Taking ln of both sides

ln1 = lne^t

t = ln1

t = 0

Substituting t = 0 into the slope formula:

m = 2(t-1)/e^t

m = 2(0-1)/e^0

m = 2(-1)/1

m = -2/1

m = -2

Substituting the slope m= -2 and the given point (1,1) into the formula for calculating the equation of tangent line we will have;

y-1 = -2(x-1)

y-1 = -2x+2

y+2x = 2+1

y+2x = 3

Hence the equation of the tangent line in Cartesian coordinates for the given parameter t is y+2x = 3

pH = f(x) = -log₁₀x

1. Graphs

I used Excel to calculate the pH values and draw the graphs (see the Figure).

f(x) and f(x) +1 are plotted against the left-hand axis, while f(x+ 1) is plotted against the expanded right-hand axis.

The points at which pH = 0 and pH = 1 are indicated by the large red dots.

2. x = 0.5

When x = 0.5, pH ≈0.30. The point is indicated by the red diamond.

3. Transformations

(a) ƒ(x) = -log(x) + 1

This function has no y-intercept, because log(0) is undefined.

(b) ƒ(x +1) = -log(x + 1)

f(0) = -log(0 + 1) = -log(1) = 0

This function has a y-intercept at (0,0).

hope this helps please mark me brainliest!