1.
a) For the Function and point below , Find f’(a).
b) Determine the equation of the line tangent to the graph of f at (a,f(a)) for the given value of
f(x) = 4x2+2x, a =1
F’(a) =
y =
2.
For the function find f’ using the definition f’(x) = lim(h→0)
Determine the equation of the line tangent to the graph of f at (a,f(a)) for the given value of
F(x) = √x+2, a=2
f’(x) =
y =
3.
find the derivative of the following function by expanding the expression
g(r) = (9r3+2r+5)(r2+6)
1a) To find f'(a), we need to take the derivative of the given function f(x) = 4x^2 + 2x and evaluate it at x = a.
The derivative of f(x) is given by f'(x) = 8x + 2.
Hence, f'(a) = 8a + 2.
1b) To determine the equation of the line tangent to the graph of f at (a, f(a)), we can use the point-slope form of a line.
The slope of the tangent line is equal to the derivative of f at the point (a, f(a)), which we found to be f'(a) = 8a + 2.
So, the equation of the tangent line is y - f(a) = f'(a) * (x - a).
Substituting the values f(a) = 4a^2 + 2a and f'(a) = 8a + 2, we get:
y - (4a^2 + 2a) = (8a + 2) * (x - a).
Simplifying this equation will give us the final equation of the tangent line.
2a) To find f'(x) using the definition f'(x) = lim(h→0) [f(x + h) - f(x)] / h, we substitute the given function f(x) = √x + 2 into the definition.
We have:
f'(x) = lim(h→0) [(√(x + h) + 2) - (√x + 2)] / h.
Solving this limit will give us f'(x).
2b) To find the equation of the tangent line to the graph of f at (a, f(a)), we will first evaluate f'(x) at x = a to get f'(a).
Then, we can use the point-slope form of a line to determine the equation of the tangent line.
The equation will be y - f(a) = f'(a) * (x - a).
Substituting the values f(a) and f'(a), we can simplify the equation to get the final result.
3) To find the derivative of the function g(r) = (9r^3 + 2r + 5)(r^2 + 6), we can use the product rule.
The product rule states that if f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x).
For this problem, let's find the derivatives of (9r^3 + 2r + 5) and (r^2 + 6) individually.
The derivative of (9r^3 + 2r + 5) with respect to r will give us g'(r).
The derivative of (r^2 + 6) with respect to r will give us h'(r).
Then, we can use the product rule formula to calculate g'(r) * (r^2 + 6) + (9r^3 + 2r + 5) * h'(r).
Simplifying this expression will give us the derivative of g(r).