Question

Suppose that $f(2+h)-f(2)=3 h^{2}+5 h .$ Calculate: (a) The slope of the secant line through $(2, f(2))$ and $(6, f(6))$ (b) $f^{\prime}(2)$

Name: Problem 24, Chapter 3: Calculus: Early Transcendentals
Uploaded: 2019-06-03T14:55:38-08:00
Duration: 2 min 25 s
Channel: Bahar Tehranipoor
Description: Suppose that f(2+h)-f(2)=3 h^2+5 h . Calculate: (a) The slope of the secant line through (2, f(2)) and (6, f(6)) (b) f^'(2)

   Suppose that $f(2+h)-f(2)=3 h^{2}+5 h .$ Calculate:
(a) The slope of the secant line through $(2, f(2))$ and $(6, f(6))$
(b) $f^{\prime}(2)$

Calculus: Early Transcendentals

Jon Rogawski, Colin… 4th Edition

Chapter 3, Problem 24

Instant Answer

Solved by Expert Bahar Tehranipoor

06/03/2019

Step 1

Now, we can rewrite $f(6)$ as $f(2 + 4)$, and use the given expression for $f(2+h) - f(2)$: $f(6) - f(2) = f(2 + 4) - f(2) = 3(4)^2 + 5(4) = 48 + 20 = 68$. So, the slope of the secant line is: $\frac{f(6) - f(2)}{4} = \frac{68}{4} = 17$. (b) Show more…

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Suppose that $f(2+h)-f(2)=3 h^{2}+5 h .$ Calculate: (a) The slope of the secant line through $(2, f(2))$ and $(6, f(6))$ (b) $f^{\prime}(2)$

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Key Concepts

Secant Line

The secant line connects two distinct points on a function's graph and its slope represents the average rate of change of the function between these points. In calculus, it gives an approximation to the instantaneous rate of change when the interval between the points is finite.

Difference Quotient

The difference quotient is the expression (f(a + h) - f(a)) / h, which calculates the average rate of change of the function over an interval of length h. It serves as a foundation for defining the derivative when taking the limit as h approaches zero.

Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as h approaches zero. This limit, when it exists, gives the instantaneous rate of change of the function at that point, offering critical insight into its local behavior.

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