Question

Find an equation of the line tangent to the following curve at the point (a,f(a)).

f(x)=7e^x +8x; a=0

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the equation of the tangent line at the point (a, f(a)), we will need to find the derivative of the function at x = a.

First, let's find the derivative of f(x):

f(x) = 7e^x + 8x

f'(x) = 7e^x + 8

Next, let's find the derivative at x = a:

f'(a) = 7e^a + 8

Now, we can write the equation of the tangent line using the point-slope form:

y - f(a) = f'(a)(x - a)

Substituting in the values:

y - f(a) = (7e^a + 8)(x - a)

Since a = 0, we have:

y - f(0) = (7e^0 + 8)(x - 0)

Simplifying:

y - f(0) = (7 + 8)(x - 0)

The constant term f(0) can be found by substituting x = 0 into the original function:

f(0) = 7e^0 + 8(0) = 7

So the equation of the tangent line is:

y - 7 = 15x