Calculus Derivatives Study Guide – Flashcards
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Limit Definition
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d/dx = lim(∆x→0) f(∆x+x) - f(x) / ∆x
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Finding points where tangent line to function is horizontal. (Horizontal tangent line to a function)
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1. Take derivative 2. Derivative=0 3. Solve for x 4. Plug x into original for y
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Equation of line normal to function at given point.
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1. Take derivative 2. Plug given x value into derivative (for slope) 3. Negative reciprocal of slope 4. Plug into point-slope form
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Evaluating Trig Functions: Sin
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sin0 = 0 sin(π/6) = 1/2 sin(π/4) = √2/2 sin(π/3) = √3/2 sin(π/2) = 1
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Evaluating Trig Functions: Cos
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cos0 = 1 cos(π/6) = √3/2 cos(π/4) = √2/2 cos(π/3) = 1/2 cos(π/2) = 0
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Evaluating Trig Functions: Tan
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tan0 = 0 tan(π/6) = √3/3 tan(π/4) = 1 tan(π/3) = √3 tan(π/2) = U
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Evaluating Trig Functions: Csc
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csc0 = U csc(π/6) = 2 csc(π/4) = √2 csc(π/3) = 2/√3 csc(π/2) = 1
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Evaluating Trig Functions: Sec
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sec0 = 1 sec(π/6) = 2/√3 sec(π/4) = √2 sec(π/3) = 2 sec(π/2) = U
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Evaluating Trig Functions: Cot
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cot0 = U cot(π/6) = √3 cot(π/4) = 1 cot(π/3) = √3/3 cot(π/2) = 0
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Find value of k so function f(x) is tangent to a line.
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1. Find derivatives of both equations 2. Set derivatives = (slopes are =) 3. Set originals = (graphs intersect) 4. Decide which equation is easier to solve for either k 5. Substitute k value into other equation (only x left) 6. Solve for x 7. Plug x value into equation to find k
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Functions aren't differential if...
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-Vertical tangent lines -Sharp turns
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Derivative of: sin(x)
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cos(x)
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Derivative of: cos(x)
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-sin(x)
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Derivative of: tan(x)
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sec²(x)
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Derivative of: sec(x)
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sec(x)tan(x)
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Derivative of: cot(x)
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-csc²(x)
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Derivative of: csc(x)
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-csc(x)cot(x)
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Finding d²y/dx²
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1. Solve for dy/dx 2. Take derivative of dy/dx 3. Plug in dy/dx into derivative of dy/dx (possible fraction buster)
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Average rate of change/average velocity
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f(b)-f(a) / b-a
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Instantaneous rate of change/instantaneous velocity
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f'(x)
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What values does average velocity = instantaneous velocity.
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1. Set avg. & inst. equations = 2. Plug in equations as Y1 & Y2 into graphing calculator 3. Set window using given interval 4. Graph → calc → intersections 5. Make sure x values are within interval
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Volume of Sphere
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V = (4/3)πr³
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Surface Area of Sphere
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SA = 4πr²
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Volume of Right Circular Cylinder (& General Prisms)
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V = Bh
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Volume of Square Pyramid
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V = (1/3)Bh
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Volume of Right Circular Cone
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V = (1/3)πr²h
