4. The point P (0.5, 0) lies on the curve y = coS TX a. If Q is the point (x, Ta), use your calculator to find the slope of the secant line PQ COS (correct to six decimal places) for the following values of x: i. 0 ii. 0.4 iii. 0.49 iv. 0.499 V. 1 vi. 0.6 vii. 0.51 viii. 0.501 b. Using the results of part (a), guess the value of the slope of the tangent line to the curve at P (0.5, 0). c. Using the slope from part (b), find an equation of the tangent line to the curve at P (0.5, 0) d. Sketch the curve, two of the secant lines, and the tangent line.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
On 4d, I calculated my secant line to be y=-2x+1 so those are my two secant lines. As for the tangent line, I am a little bit confused on how to calculate it. I put up this question already and got an answer, but I thought tangent lines couldn't go through the graph.
Please tell me if I calculated the secant line correctly and how to calculate the tangent line (show work).
Thank you!
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