1.4 Rates of Change and Tangent Lines

given piecewise function and finding the left and right limit: even if they are the same, what do you have to do? the limit approaches? If the left and right slopes are different, then their is no? remember: the left and right limits must be the same for there to be a? Two conditons for these problems to have a tangent line? (2)

use the quotient equation, x =a, tangent line, tangent line at x =a,left and right limits must be the same/slopes of both must be the same,

if m =0 for the normal or tangent line: the line is a? equation? ex. x = 4 of (4, -2)

vertical line, x = x of the point,

given two pairs of values: equation for slope of curve with 2 points? then, plug in x and y values to find the?

y2-y1/x2-x1, b intercept

to find the rate of change of the area of a circle with a r =9: first, what is the average rate of change equation with 2 points? what is equation when there is only 1 given point? given problem f(x) = x^2 sin (1/x), x cannot equal 0 and 0, x =0: to determine if it has a tangent, follow these 3 steps? (3) ex. slope of x^2 sin (1/x) is? slope of x= 0 is?

y2-y1/x2-x1, lim h approaches 0 f(h+a)-f(a)/h,use squeeze theorem/check the left and right limits of the inequality/if they are the sams, then a limit and a slope exists, 0, 0

Average rate of change is? Ex. F(x) = x^3-x, [1,3]. To solve, find? (2) equation? The answer is 24-0/2 = 12. Secant to a curve line def.? The slope of the secant line =?

Change/time, f(1) / f(3), y2-y1/x2-x1, a line through two pots on a curve average rate of change,

Difference quotient of f at A equation? The limit is equal to the? The slope and limit are also equal to the? The Normal line to a curve at a point is the line that is? Ex. Find the normal line to the curve given f(x) = -x^2+4. Use which equation? Find the? Then, for the perpendicular slope, find the? Ex. M = -2. Perpendicular slope? Use which prior points to create the perpendicular linear equation? (2)

F(a+h) -f(a)/h, secant slope, average rate of change, perpendicular, slope of a curve or secant equation, negative reciprocal, 1/2, (a, f(a) ),

Instantaneous rate of change is the same as using the? Free fall uses the?

Quotient equation, quotient equation

Instantaneous rate of change is the same as using the? Free fall uses the? To find the secant slope: equation? To find the velocity, look for the value as? Steps to find normal of the curve? (3) for the equation, use?-equation? Remember: in the slope of a tangent or secant and quotient equation, h equation is? Ex. h+a for y = -x^2 at x=-2 is?

Quotient equation, quotient equation, y2-y1/x2-x1, Q approaches P,find the slope of the tangent line/find the perpendicular slope/create the perpendicular equation, point slope form, y-y2 = m (x-x1),h +a, h-2

The slopes of the secants approach the? (Of the?) Finding the tangent line: what do you do? (3) ex. Y = x^2, P = (2, 4). Equation for P point? Equation for Q point? Plug h+a into? Overall slope of a curve or secant equation? Write the limit as? After you find the slope, create a? Use which point values to create it?

Slope of the tangent, find the slope of the secant through points P and Q/find the lim secant slope as Q approaches P/find the slope and the linear equation, (a, f(a) ), (h+a), f(h+a), f(x), m = lim h approaches 0 f(h+a)-f(a)/h, lim Q approaches P = lim h approaches 0, linear equation, P point,

To find the tangent to a curve: first, use the? For h+a, use? Find the? Make it Equal to the? Find the? Plug each one back into the? Find the? For each x value, create a? Use which form for the equation?

Slope or quotient equation, h+x, m equation, given m value, x values, original function, y values, tangent equation, point slope form

To find the points (x,y) at which f(x) has a horizontal tangent: first, use the? For h +a, substitute? Solve to find the? Then Solve that for the? To find the y value, plug the x back into the?

Slope or quotient equation, h+x, slope equation, x value, original function

Given f(x) = x^2 sin 1/x, x cannot equal 0, and 0 x = 0, what do you do to find the tangent line and slope? (3) the limit must equal the slope for there to be a? Lim, slope, and derivative equation? Derivative and slope and limit equation?

Use the squeeze theorem to find the lim of the function/find the slope of the second function/check to see if they are the same, tangent line, lim x approaches c f(x) = m of tangent = derivative, lim h approaches 0 f(h+a)-f(a)/h

Given y = -2x^2 and x = a, plug in for the slope of a tangent equation...? For the f(a) portion, plug in? Subtract the? Keep track of the? (2) make sure to distribute the? Given a final equation of -4/(a-1)^2, the slope is always? The slope is very steep at which point?

h +a, a, original function, +\- signs, -sign, negative, a = 1

Slope of a curve or secant equation? The tangent line to the curve at P is the? F(x) = 1/x, find the slope at a. Use which formula? Solve them -1/a^2 = -1/4. Two answers? Plug these into the original equation? The new points are? (2)

m = lim h approaches 0 f(h+a)-f(a)/h, line through P, slope of a curve or secant equation, +_/2, 1/a, (a, f(a) ) / (-a, (a) ),

Given: find the slope of the absolute value f(x) = |4x| ay x = 3 and x = -4. For the positive x, use the? For the negative x, use the?-ex.? Given f(x) = 5-5x-x^2 x <0 and -2x + 5, x>_ 0, what do you do to find the slope of the tangent line? (2) when the x is < or <_ a value, then it is the? When x is > or >_ a value, it is the?

positive absolute value, negative absolute value, y = -4x, fine the left and right limit/check if they are the same, left limit, right limit

remember: the instantaneous speed is always? ex. m = -19.6 ft/sec is actually? finding the points of the tangent that make it horizontal: solve with the? set m =? solve for? plug x back into the? Given P and points Q1, Q2, etc., find the average change of? (2)

positive, 19.6 ft/sec, quotient equation, 0, x, original equation, y2-y1/x2-x1/ P- point Q

absolute value functions and finding slope of the curve: given |x-2| and x =2 and x=3 and x = -1: absolute value functions have two forms which are? (2) ex. since x=2 and x=3 aee >_ 2, the absolute value form is then? ex. x-2. since x =-1 is < 2, the absolute value form you use is? ex. -x-2. another function |x|: for which x values do you use the positive form? negative?

positive/negative, positive, negative, >_0, <0,

given square root 5h +25 -5/h: how do you solve? conjugate is? multiply thid by both the? for y = -2tan x at (-pie, 0) and find the tangent slope: to solve, you can use a? (2)find the lim f(x) as x...?

simplify the numerator, square root 5h +25 +5, numerator and denominator, graph/table of values, approaches 0,

not likely to be on the quiz: given 2 secant points, use y2-y1/x2-x1 to find the? then plug in points to find the? to find the equation of the tangent, choose the? solve with the? plug in points to find the?

slope of the secant line, equation, leftmost x value, slope of the curve equation, equation