calculus
posted by Adam Janky .
Consider the following.
y = (2x2 + 5)(x3 − 25x)
at (5, 0)
(a) At the indicated point, find the slope of the tangent line.
(b) Find the instantaneous rate of change of the function.

calculus 
MathMate
First check if (5,0) is on the line:
f(x)= (2x2 + 5)(x3 − 25x)
f(5)= (50+5)(125125)=0 indeed.
a)Slope
Slope of the tangent line at (5,0) is the value of f'(x) at (5,0), namely f'(5).
So let's find f'(x).
We can find f'(x) as a product, or as an expanded polynomial. I choose the latter:
y = (2x2 + 5)(x3 − 25x)
=2x^5+5x^350x^3125x^2
=2x^545x^3125x^2
Differentiate term by term:
f'(x)=10x^4135x2250x
so
f'(5)=625033751250=1625
= slope of tangent at (5,0)
b.
instantaneous rate of change of the function is precisely f'(x)=dy/dx.
So the answer is the same as in a.
Respond to this Question
Similar Questions

calculus
Write the equation of the tangent line to the curve at the indicated point. As a check, graph both the function and the tangent line. f(x) = x7 − 7 at x = −1 7 x7 
calculus
At the indicated point for the function, find the following. A graphing utility's numerical derivative feature can be used to check your work. y = (x3 + 2x)3 at x = 3 (a) Find the slope of the tangent line. (b) Find the instantaneous … 
calculus
Consider the following. y = x2 − 9x x2 + 5x at (3, − 3 4 ) (a) At the indicated point, find the slope of the tangent line. (b) At the indicated point, find the instantaneous rate of change of the function. 
calculus
i'm not sure how to do this. can someone help, please? 
calculus
To find the equation of a line, we need the slope of the line and a point on the line. Since we are requested to find the equation of the tangent line at the point (64, 8), we know that (64, 8) is a point on the line. So we just need … 
calculus
For the function y = 9x2 + 6x + 2, at the point x = 7, find the following. (a) the slope of the tangent to the curve (b) the instantaneous rate of change of the function 
calculus
please anyone help me out Consider the following. Function Point f(x) = 5x^2 − 4, (3, 41) Find an equation of the tangent line to the function at the given point. y = Find the function values and the tangent line values at f(x … 
Calculus
Use the fourstep process to find the slope of the tangent line to the graph of the function at the given point and determine an equation of the tangent line. f(x) = 2x22x+1 (1, 2) 
calculus
1. Which of the following expressions is the definition of the derivative of f(x) = cos(x) at the point (2, cos(2))? 
calculus
8. Find the derivative of the function f(x) = cos^2(x) + tan^2(x) 9. Find f '(x) for f(x) = xln(x)  5ex. 10. Find f '(x) for f (x) = e(^lnx) + e^(x)ln(x) 11. Find the xcoordinate where the graph of the function f(x) = e^(cosx) has …