Use the limit process to find the slope of the graph of the function at the specified point. Use a graphing utility to confirm your result.f(x) = 2x − 6x2, (3, −48)
Question
Use the limit process to find the slope of the graph of the function at the specified point. Use a graphing utility to confirm your result.f(x) = 2x − 6x2, (3, −48)
Solution
To find the slope of the graph of the function at a specified point using the limit process, we need to find the derivative of the function. The derivative of a function at a certain point gives the slope of the tangent line at that point.
The function is f(x) = 2x - 6x^2.
The derivative of a function f(x) is given by the limit:
f'(x) = lim(h->0) [f(x+h) - f(x)] / h
Let's find the derivative of f(x):
f'(x) = lim(h->0) [(2(x+h) - 6(x+h)^2) - (2x - 6x^2)] / h
Simplify the expression inside the brackets:
= lim(h->0) [(2x + 2h - 6x^2 - 6h^2) - (2x - 6x^2)] / h
= lim(h->0) [2h - 6h^2] / h
Factor out h:
= lim(h->0) h(2 - 6h) / h
Cancel out h:
= lim(h->0) 2 - 6h
As h approaches 0, the limit is 2.
So, the derivative of the function f(x) = 2x - 6x^2 is f'(x) = 2.
The slope of the graph of the function at the point (3, -48) is therefore f'(3) = 2.
You can confirm this result using a graphing utility by graphing the function and its derivative and checking the slope at the point (3, -48).
Similar Questions
Find the points on the graph of the function that are closest to the given point.f(x) = x2, (0, 6)
For the function f whose graph is shown, state the following. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.)FigureThe x y coordinate plane is given. Refer to the adjacent description for more details.Description(a)lim x → −7 f(x)(0,2) (b)lim x → −3 f(x)DNE (c)lim x → 0 f(x)2 (d)lim x → 6− f(x)∞ (e)lim x → 6+ f(x)−∞ (f)the equations of the vertical asymptotes (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)x = (−3,6)
Fill in the blank.The of a function f at x represents the slope of the graph of f at the point (x, f(x)).
Sketch the graph of a function f that satisfies all of the given conditions.lim x→6+ f(x) = 7, lim x→6− f(x) = 5, lim x→−2 f(x) = 5, f(6) = 6, f(−2) = 4
The function f is defined for all x∈ℝ. The line with equation y=6x-1 is the tangent to the graph of f at x=4.The function g is defined for all x∈ℝ where g(x)=x2-3x and h(x)=f(g(x)).Write down the value of f′(4).
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.