Derivatives of Polar Functions
Derivatives of polar functions involve finding the rate of change of a polar curve with respect to the angle. This is done using the polar form of the chain rule and involves expressing the polar coordinates in terms of the Cartesian coordinates. The derivative of a polar function gives the slope of the tangent line to the curve at a specific point.
3 Key excerpts on "Derivatives of Polar Functions"
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...In words, the numerator of this formula represents the second derivative of y (t) and the denominator represents the first derivative of x (t). Since The most common mistake in finding the second derivative is to use the quotient rule to find the derivative of the first derivative. It does not work that way! O * Derivatives of polar equations 1. Rewrite the polar equations in parametric form and use the parametric formulas. i. First derivative—rewrite r = f (θ) in parametric find the derivative (slope) of f (θ) = 2 sin(3 θ) at Graphically, this represents the slope of the tangent line to the graph of f (θ) = 2 sin (3 θ) at the point. ii. Second derivative of a polar equation is found using the parametric formula for the second derivative, though it does not appear on the AP Calculus exams. Keep in Mind... Whens using the quotient rule, do not switch the order of the terms in the numerator since subtraction is not commutative. When differentiating a function of the form where k is a constant, do not use the quotient rule. The derivative is simply since is a constant that can be factored out. When asked to find the derivative of parametric equations at a certain point, pay attention to whether you are given an x -value or a t -value and solve the problem accordingly. The second derivative of parametric equations is tricky; make sure you don’t fall for it! Practice it until you get it right. Do not confuse ln(1) = 0 with ln(0) = 1. The former is true since x = 1 is the x -intercept of y = ln(x). The latter is false since x = 0 is not in the domain of y = ln(x). Don’t forget to use the product rule in implicit differentiation problems in which you must take the derivative of a product involving both x and y...
eBook - ePub- Richard C. Dorf, Ronald J. Tallarida(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...6 Differential Calculus 1. Notation For the following equations, the symbols f (x), g (x), etc., represent functions of x. The value of a function f (x) at x = a is denoted f (a). For the function y = f (x) the derivative of y with respect to x is denoted by one of the following: d y d x, f ′ (x), D x y, y ′. Higher derivatives are as. follows: d 2 y d x 2 = d d x (d d x) = d d x f ′ (x) = f ″ (x) d 2 y d x 3 = d d x (d 2 y d x 2) = d d x f ″ (x) = f ′ ″ (x), etc. and values of these at x = a are. denoted f ″ (a), f ‴ (a), etc. (see Table of Derivatives). 2. Slope of a Curve The tangent line at a point P (x, y) of the curve y = f (x) has a slope f ′(x) provided that f ′(x) exists at P. The slope at P is defined to be that of the tangent line at P. The tangent line at P (x 1, y 1) is given by y − y 1 = f ′ (x 1) (x − x 1). The normal line to the curve at P (x 1, y 1) has slope −1/ f ′(x 1) and thus obeys the equation y − y 1 = [ − 1 / f ′ (x 1) ] (x − x 1) (The slope of a vertical line is not defined.) 3. Angle of Intersection of Two Curves Two curves, y = f 1 (x). and y = f 2 (x), that intersect at a point P (X, Y) where derivatives f ′ 1 (X), f ′ 2 (X) exist, have an angle (α) of intersection given by tan α = f ′ 2 (X) − f ′ 1 (X) 1 + f ′ 2 (X) ⋅ f ′ 1 (X) ⋅ If tan α > 0, then α is the acute angle; if tan α < 0, then α is the obtuse angle. 4. Radius of Curvature The radius of curvature R of the. curve y = f (x) at point P (x, y) is R = { 1 + [ f ′ (x) ] 2 } 3 / 2 f ″ (x) In polar coordinates (θ, r) the corresponding formula is R = [ r 2 + (d r d θ) 2 ] 3 / 2 r 2 + 2 (d r d θ) 2 − r d 2 r d θ 2 The curvature K is 1/ R. 5. Relative Maxima and. Minima The function f has a relative maximum at x = a if f (a) ≥ f (a + c) for all values of c (positive or negative) that are sufficiently near zero. The function f has a relative minimum at x = b if f (b) ≤ f (b + c) for all values of c that are sufficiently close to zero...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...The term requires using the quotient rule. Multiply each term by y 2. The point-slope form of the equation is The implicit curve and the line tangent at (1, 1) are shown in Figure 4.2. Figure 4.2 4.9 DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS Implicit differentiation enables the exploration of the derivatives of an additional set of functions, inverse trigonometric functions. As with the normal trigonometric functions, there are six inverses. The first derivative will be justified, and the rest will be presented without proof. Once again, these derivatives need to be committed to memory, but fortunately, they have cofunction patterns that will ease the task. Remember, the domain values of the inverse trigonometric functions are trigonometric ratios, real numbers, and the range consists of radian measures with appropriately defined limitations. The function y = arctan(x) essentially says, “Tell me the angley with a tangent value of x.” (Note: Unless otherwise noted, arctan(x), will be also written with the notation tan –1 (x). This notation is also used with the other inverse functions, and should not be thought of as a power of – 1.) for y = tan –1 (x) denotes the instantaneous rate of change of the radian measure as the trigonometric ratio changes value. It also finds an expression for the slope of the graph of the inverse function at any given point of its domain. To find for y = tan –1 (x), we write an equivalent expression, x = tan(y). The implicit differentiation of x = tan(y) follows. Unfortunately, since y is originally a function of x, the derivative also should be in terms of x. This is where a bit of right triangle trigonometry is helpful! Figure 4.3 Figure 4.3 shows a right triangle with an acute angle y and the legs labeled appropriately so that tan(y) = x...
