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1. Set up the difference quotient for the following derivatives, then calculate the limit without using any derivative rules. a. f(x)=√x + 2, at a = 7. b. s(t) = 2 -, at a = x. 3-t c. h(x) = x² + 2x + 5, at a = − 1.

1. Set up the difference quotient for the following derivatives, then calculate the limit without using any derivative rules. a. f(x)=√x + 2, at a = 7. b. s(t) = 2 -, at a = x. 3-t c. h(x) = x² + 2x + 5, at a = − 1.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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1. Set up the difference quotient for the following derivatives, then calculate the limit without using any derivative rules. a. f(x)=√x + 2, at a = 7. b. s(t) = 2 -, at a = x. 3-t c. h(x) = x² + 2x + 5, at a = − 1.
Transcribed Image Text:1. Set up the difference quotient for the following derivatives, then calculate the limit without using any derivative rules. a. f(x)=√x + 2, at a = 7. b. s(t) = 2 -, at a = x. 3-t c. h(x) = x² + 2x + 5, at a = − 1.
MULTIPLE DERIVATIVES A function fis-times differentiable (or fE C") if you can apply the derivative times to fand have a continuous function after each application of the derivative. A function fis smooth (orf E C) if it can be differentiated infinitely many times, and each derivative is a continuous function Linearity: Products: Quotients: Compositions: DERIVATIVE PROPERTIES d [f(x) + a· g(x)] = f(x) + a · g(x) dx d dx [ƒ(x)g (x)] = f(x)g (x) + f(x)g'(x) d f(x) dx [g(x)] d dx f(x)g(x) = f(x)g'(x) (g(x)) ² where g(x) = 0 [ƒ (g(x))] = f(g(x)) · g'(x) TANGENT AND NORMAL LINES If y=f(x) describes some differentiable function, the equation of the tangent line at a point is given by y = f(x)(x − a) + f(a). The equation of the normal line at a point (x-a) + f(a). If fis differentiable near f (a) is given by LINEAR APPROXIMATION then for values close to, f(x) = f(a)(x-a) + f(a). Constant: Power: Exponential: Logarithmic: BASIC FUNCTION DERIVATIVES Trigonometric: Hyperbolic: d dx d dx d dx d dx d dx d d [b] = ln(b) b*, where b € (0,0) dx d dx [a] = 0, where a d dx M dx d dx d dx Inverse Trigonometric: d dx d dx d dx =x²-¹, where [log(x)] = [sin(x)] = cos(x) [cos(x)]=sin(x) [tan(x)] = sec²(x) [sec(x)] = sec(x)tan(x) [cot(x)]=-csc²(x) [csc(x)]=csc(x)cot(x) [arcsin(x)] [arccos(x)] In(b) x [arctan(x)] 1+x² where b, x € (0,00) [sinh(x)] = cosh(x) [cosh(x)] = sinh(x) where x = ± where x ± C ZOOM +
Transcribed Image Text:MULTIPLE DERIVATIVES A function fis-times differentiable (or fE C") if you can apply the derivative times to fand have a continuous function after each application of the derivative. A function fis smooth (orf E C) if it can be differentiated infinitely many times, and each derivative is a continuous function Linearity: Products: Quotients: Compositions: DERIVATIVE PROPERTIES d [f(x) + a· g(x)] = f(x) + a · g(x) dx d dx [ƒ(x)g (x)] = f(x)g (x) + f(x)g'(x) d f(x) dx [g(x)] d dx f(x)g(x) = f(x)g'(x) (g(x)) ² where g(x) = 0 [ƒ (g(x))] = f(g(x)) · g'(x) TANGENT AND NORMAL LINES If y=f(x) describes some differentiable function, the equation of the tangent line at a point is given by y = f(x)(x − a) + f(a). The equation of the normal line at a point (x-a) + f(a). If fis differentiable near f (a) is given by LINEAR APPROXIMATION then for values close to, f(x) = f(a)(x-a) + f(a). Constant: Power: Exponential: Logarithmic: BASIC FUNCTION DERIVATIVES Trigonometric: Hyperbolic: d dx d dx d dx d dx d dx d d [b] = ln(b) b*, where b € (0,0) dx d dx [a] = 0, where a d dx M dx d dx d dx Inverse Trigonometric: d dx d dx d dx =x²-¹, where [log(x)] = [sin(x)] = cos(x) [cos(x)]=sin(x) [tan(x)] = sec²(x) [sec(x)] = sec(x)tan(x) [cot(x)]=-csc²(x) [csc(x)]=csc(x)cot(x) [arcsin(x)] [arccos(x)] In(b) x [arctan(x)] 1+x² where b, x € (0,00) [sinh(x)] = cosh(x) [cosh(x)] = sinh(x) where x = ± where x ± C ZOOM +
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