Part 1: The derivative at a specific point Use the definition of the derivative to compute the derivative of f(x) = 5/x at the specific point x = 2. Evaluate the limit by using algebra to simplify the difference quotient (in first answer box) and then evaluating the limit (in the second answer box). f'(2) = lim_{h -> 0} ( (f(2 + h) - f(2)) / h ) = lim_{h -> 0} ( (-5) / (2(2 + h)) ) = -5/4. Part 2: The derivative function Use the definition of the derivative to compute the derivative of the function f(x) = 5/x at an arbitrary point x. Evaluate the limit by using algebra to simplify the difference quotient (in first answer box) and then evaluating the limit (in the second answer box). f'(x) = lim_{h -> 0} ( (f(x + h) - f(x)) / h ) = lim_{h -> 0} ( ) = . Part 3: The tangent line
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To find \( f'(2) \), we use the definition of the derivative: \[ f'(2) = \lim_{h \to 0} \left( \frac{f(2 + h) - f(2)}{h} \right) \] First, compute \( f(2 + h) \) and \( f(2) \): \[ f(2 + h) = \frac{5}{2 + h} \] \[ f(2) = \frac{5}{2} \] Substitute these into the Show more…
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The derivative at a specific point Use the definition of the derivative to compute the derivative of f(x) at the specific point x = a by finding the difference quotient (in the first answer box) and then evaluating the limit (in the second answer box) lim(x→a) [f(x + h) - f(x)]/h. Part 2: The derivative function Evaluate the limit by using algebra to simplify the difference quotient (in the first answer box) and then evaluating the limit (in the second answer box) lim(h→0) [f(a + h) - f(a)]/h. Part 3: The tangent line Now let's calculate the tangent line to the function f(x) = at x = 5. a. By using f'(x) from part 2, the slope of the tangent line to f at x = 5 is f'(5). b. The tangent line to f at x = 5 passes through the point (5, f(5)) on the graph of f. (Enter point in the form (2, 3) including the parentheses.) c. An equation for the tangent line to f at x = 5 is y = f'(5)(x - 5) + f(5).
Brent B.
Part 1: Limit of a difference quotient Suppose f(x) Evaluate the limit by using algebra to simplify the difference quotient (in first answer box) and then evaluating the limit (in the second answer box): lim f(5+h) - f(5) / h = lim Part 2: Interpreting the limit of a difference quotient
Khushbu R.
Part 1: Limit of a difference quotient Suppose f(x) = 7x^2 + 8x - 3. Evaluate the limit by using algebra to simplify the difference quotient (in first answer box) and then evaluating the limit (in the second answer box). Part 2: Interpreting the limit of a difference quotient The limit of the difference quotient, -20, from Part 1 above is (select all that apply). A. the average rate of change of f at x = -2. B. the slope of the tangent line to the graph of y = f(x) at x = -2. C. the instantaneous rate of change of f at x = -2. D. f'(-2) E. f(-2). F. the slope of the secant line to the graph of y = f(x) at x = -2.
Avinash V.
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