Explanation For a graph of the function f , difference quotient computes the slope of a secant line through two points on the graph. The x -coordinates of these two points are x and x + h . Given: f ( x ) = x 5 / 2 Formula Used: The difference quotient is: f ( a + h ) − f ( a ) h f ' ( a ) = lim h → 0 f ( a + h ) − f ( a ) h Calculation: The graph shows that the given function is an increasing function. It is evident from the graph that the slope of the secant line at x = 4 − h is less than the slope of the tangent line at x = 4 which is less than the slope of the secant line at x = 4 + h . (a) For smaller value of h = 0.0001 f ( 4 ) − f ( 3.999 ) 0.0001 = 0.019996 0.0001 = 19.996 f ' ( 4 ) = 20 f ( 4 + h ) − f ( 4 ) h = 0.00200004 0.0001 = 20...

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

3rd Edition

ISBN: 9781319048518

Author: Colin Adams Jon Rogawski

Publisher: Macmillan Higher Education

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Question

Chapter 3.1, Problem 63E

To determine

We need to plot the graph of $f (x) = x^{5 / 2}$ on $[0, 6]$

(a) We need to justify the inequalities for $h > 0$

$\frac{f (4 - h) - f (4)}{h} \leq f^{'} (4) \leq \frac{f (4 + h) - f (4)}{h}$

(b) We need to compute $f^{'} (4)$

(c) We need to plot $y = f (x)$ and the tangent line at $x = 4$

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Chapter 3.1, Problem 62E

Chapter 3.1, Problem 64E

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The OU process studied in the previous problem is a common model for interest rates. Another common model is the CIR model, which solves the SDE: dX₁ = (a = X₁) dt + σ √X+dWt, - under the condition Xoxo. We cannot solve this SDE explicitly. = (a) Use the Brownian trajectory simulated in part (a) of Problem 1, and the Euler scheme to simulate a trajectory of the CIR process. On a graph, represent both the trajectory of the OU process and the trajectory of the CIR process for the same Brownian path. (b) Repeat the simulation of the CIR process above M times (M large), for a large value of T, and use the result to estimate the long-term expectation and variance of the CIR process. How do they compare to the ones of the OU process? Numerical application: T = 10, N = 500, a = 0.04, x0 = 0.05, σ = 0.01, M = 1000. 1 (c) If you use larger values than above for the parameters, such as the ones in Problem 1, you may encounter errors when implementing the Euler scheme for CIR. Explain why.

#8 (a) Find the equation of the tangent line to y = √x+3 at x=6 (b) Find the differential dy at y = √x +3 and evaluate it for x=6 and dx = 0.3

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Chapter 3 Solutions

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

Ch. 3.1 - Prob. 1PQ Ch. 3.1 - Prob. 2PQ Ch. 3.1 - Prob. 3PQ Ch. 3.1 - Prob. 4PQ Ch. 3.1 - Prob. 5PQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E

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We need to plot the graph of f ( x ) = x 5 / 2 on [ 0 , 6 ] (a) We need to justify the inequalities for h &gt; 0 f ( 4 − h ) − f ( 4 ) h ≤ f ' ( 4 ) ≤ f ( 4 + h ) − f ( 4 ) h (b) We need to compute f ' ( 4 ) (c) We need to plot y = f ( x ) and the tangent line at x = 4

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We need to plot the graph of f(x)=x5/2on [0,6] (a) We need to justify the inequalities for h>0 f(4−h)−f(4)h≤f'(4)≤f(4+h)−f(4)h (b) We need to compute f'(4) (c) We need to plot y=f(x)and the tangent line at x=4

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We need to plot the graph of f ( x ) = x 5 / 2 on [ 0 , 6 ] (a) We need to justify the inequalities for h > 0 f ( 4 − h ) − f ( 4 ) h ≤ f ' ( 4 ) ≤ f ( 4 + h ) − f ( 4 ) h (b) We need to compute f ' ( 4 ) (c) We need to plot y = f ( x ) and the tangent line at x = 4

We need to plot the graph of $f (x) = x^{5 / 2}$ on $[0, 6]$

(a) We need to justify the inequalities for $h > 0$

$\frac{f (4 - h) - f (4)}{h} \leq f^{'} (4) \leq \frac{f (4 + h) - f (4)}{h}$

(b) We need to compute $f^{'} (4)$

(c) We need to plot $y = f (x)$ and the tangent line at $x = 4$