Explain what is incorrect with the following limit definition answers, and correct the errors. There is at least one error in each solution. Hint: If you cannot find errors, try working the problems yourself. (a) Using the limit definition of the derivative find the derivative f'(x) of the function f(x) = 2x² + 5x. f (x + h) = 2(x + h)² + 5(x + h) = 2x² + 2h² + 5x + 5h f (x + h) – f(x) = 2x² + 2h² + 5x + 5h – 2x² + 5x = 2h² + 10x + 5h f(x + h) – f(x) 2h2 + 10x + 5h 10x 2h + + 5 h h h f(x+ h) – f(x) 10x | lim h→0 h = 2h + + 5 = 5 h 3 (b) Using the limit definition of the derivative find the derivative f'(x) of the function f(x) 4x+5 3 3 f(x + h) = 4(x +h) + 5 4.x + 4h + 5 3 3 f(x + h) – f(x) = 4.x + 4h + 5 4.x + 5 3(4x+5)-3(4x+4h+5) (4x+5)(4x+4h+5) 3 3 -3h f (x + h) – f(x) 4x+4h+5 4x+5 (4x+5)(4x+h+5) h h h h -3h f (x + h) – f(x) (4x+5)(4x+4h+5) lim -3h? lim h→0 lim h→0 (4x + 5)(4x + 4h + 5) h h→0 h (c) Using the limit definition of the derivative find the derivative f'(x) of the function f (x) = /3x + 2. f(x + h) = v3x + h + 2 f (x + h) – f(x) = v3x + h + 2 – V3x + 2 f(r + h) – f(x) V3.x + h + 2 – V3x + 2 h h f(x + h) – f(x) V3x + h + 2 – V3x + 2 lim lim h→0 = 1 %3D h h→0 h

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: