Section 4.2 Enhanced Assignment

Instructor: Neelam Aggarwal Course: Spring 2022 - Calculus I (MATH1441-02F) Student: Jalen Wilson Date: 07/12/24 Assignment: Section 4.2 Enhanced Assignment 1. Find the value of the derivative. dy 2 dx x = − 3 dy dx x = − 3 2. if y = 4 − 9x = ​(Simplify your​answer.) 54 Find the value or values of c that satisfy the equation f(b) − f(a) b−a = f′​(c) in the conclusion of the Mean Value Theorem for the following function and interval. 2 ​f(x) = 2x − 5x − 3​, ​[ − 3​,2​] The​value(s) of c that satisfy the equation f(b) − f(a) b−a = f′​(c) is/are − 1 2 . ​(Type a simplified fraction. Use a comma to separate answers as​needed.) 3. Find the value or values of c that satisfy the equation f(b) − f(a) b−a = f′(c) in the conclusion of the Mean Value Theorem for the given function on the given interval. 8 5 f(x) = x ; [0,1] 5 c= 5 3 8 ​(Use a comma to separate answers as​needed.) 4. Find the value or values of c that satisfy the equation f(b) − f(a) b−a = f′(c) in the conclusion of the mean value theorem for the given function and interval. f(x) = 6x + c= 5. 6 x ​, 1 1 13 ,13 ​(Use a comma to separate answers as​needed.) Find the value or values of c that satisfy the equation f(b) − f(a) b−a = f′(c) in the conclusion of the mean value theorem for the function and interval shown below. ​f(x) = ln (x − 1)​, c= 6 ln 7 ​[2​,8​] +1 ​(Type an exact answer. Use a comma to separate answers as​needed.) 6. Show that the function f(x) = x4 + 6x + 1 has exactly one zero in the interval [ − 1, 0]. Which theorem can be used to determine whether a function​f(x) has any zeros in a given​interval? A. Intermediate value theorem B. ​Rolle's Theorem C. Extreme value theorem D. Mean value theorem 4 To apply this​theorem, evaluate the function f(x) = x + 6x + 1 at each endpoint of the interval [ − 1, 0]. f( − 1) = f(0) = −4 1 ​(Simplify your​answer.) ​(Simplify your​answer.) 4 According to the intermediate value​theorem, f(x) = x + 6x + 1 has at least one zero in the given interval. ​Now, determine whether there can be more than one zero in the given interval. ​ olle's Theorem states that for a function​f(x) that is continuous at every point over the closed interval [a,b] and differentiable at every point R of its interior (a,b)​, if ​f(a) = ​f(b), then there is at least one number c in (a,b) at which f′(c) = 0. 4 Find the derivative of f(x) = x + 6x + 1. f′(x) = 3 4x + 6 Can the derivative of​f(x) be zero in the interval [ − 1, 0]​? Yes No 4 The function f(x) = x + 6x + 1 has at least one zero at some point x = a in the interval [ − 1, 0]. According to​Rolle's Theorem, can there be another point x = b in this interval where f(a) = f(b) = 0? Yes No 4 ​ hus, since the intermediate value theorem shows that f(x) = x + 6x + 1 has at least one zero in the interval [ − 1, 0] and​Rolle's Theorem T shows that there cannot be two points x = a and x = b for which f(a) = f(b) in this​interval, the function​f(x) has exactly one zero in the interval [ − 1, 0]. 7. Suppose that ​f( − 3​) = 4 and that f′​(x) = 0 for all x. Must ​f(x) = 4 for all​x? Give reasons for your answer. A. Yes. Since f′​(x) = 0 for all​x, and 0 is a​constant, the value of f equals ​f( − 3​) for all values of x. B. No. Since ​f( − 3​) = 4​, f is a constant function with slope 4. The value of f is different for all values of x. C. No. Since ​f( − 3​) = 4 is greater than − 3​, ​f(x) is greater than x for all values of x. D. Yes. Since f′​(x) = 0 for all​x, f is a constant function. The value of f is the same for all values of x. 8. Suppose that f′​(x) = 2x for all x. ​ ) Find ​f( − 3​) if ​f(0) = 0. a ​b) Find ​f( − 3​) if ​f(1​) = − 3. ​c) Find ​f( − 3​) if ​f( − 4​) = 20. ​a) When ​f(0) = ​0, ​f( − 3​) = 9 . ​(Simplify your​answer.) 9. ​ ) When ​f(1​) = − 3​, ​f( − 3​) = b ​(Simplify your​answer.) 5 c​ ) When ​f( − 4​) = 20​, ​f( − 3​) = ​(Simplify your​answer.) 13 . . Find the value or values of c that satisfy the equation f(b) − f(a) b−a = f′​(c) in the conclusion of the Mean Value Theorem for the function and interval. 2 ​f(x) = x + 5x + 3​, ​[ − 3​,2​] A. ​0, − B. − 1 2 1 1 ​, 2 2 C. − 3​, 2 D. − 10. 1 2 Find the value or values of c that satisfy the equation interval. Round to the nearest thousandth. ​f(x) = ln (x − 3)​, ​[4​,6​] A. 5.820 B. 5.885 C. 4.820 D. ± 4.820 f(b) − f(a) b−a = f′​(c) in the conclusion of the Mean Value Theorem for the function and