Which of the following four fundamental funtions have the same tangent line at x = 0 Ans:_________________ ?
y = x, y = x2, y = x3 and y = ex
Tangent line at x=0 means to evaluate the derivative at x=0, or f'(0).
For the tangent lines to be equal (coincident), f(0) must also be equal.
y=x => f'(x)=1 => f'(0)=1, f(0)=0
y=x²=> f'(x)=2x => f'(0)=0, f(0)=0
y=x³ => f'(x)=3x² => f'(0)=0, f(0)=0
y=ex =>f'(x)=ex => f'(0)=e0=1, f(0)=1
Therefore there are two sets of tangent lines with equal slopes at x=0, but only one set of them have equal y-intercepts and hence coincident lines.
For:
y=x
Technically there is no tangent line to a straight line because the definition of tangent means a line a touching at one point.
Answer is:
x^2 and x^3
To determine which of the given functions have the same tangent line at x = 0, we need to find the derivatives of each function and evaluate them at x = 0.
Let's find the derivatives of each function:
1. y = x
The derivative of y = x with respect to x is simply 1, since the derivative of x with respect to x is 1.
2. y = x^2
To find the derivative of y = x^2, we can use the power rule: d/dx (x^n) = n * x^(n-1).
Applying the power rule, we get: dy/dx = 2x.
3. y = x^3
Similarly, applying the power rule to y = x^3, we get: dy/dx = 3x^2.
4. y = e^x
The derivative of y = e^x with respect to x is simply e^x, as the derivative of e^x with respect to x is e^x itself.
Now, let's evaluate the derivatives at x = 0:
1. At x = 0, dy/dx for y = x is 1.
2. At x = 0, dy/dx for y = x^2 is 2(0) = 0.
3. At x = 0, dy/dx for y = x^3 is 3(0)^2 = 0.
4. At x = 0, dy/dx for y = e^x is e^0 = 1.
Therefore, the functions y = x and y = ex have the same tangent line at x = 0 since both of their derivatives evaluate to 1 at that point.
To answer the question, either "y = x" or "y = ex" has the same tangent line as "y = x" at x = 0.