A piecewise defined function g is given by the formula -x+1 if x 4 g(x) if...

    

Transcribed Image Text:

A piecewise defined function g is given by the formula -x+1 if x 4 g(x) if x > 4 A partial graph of this function is given below. In this problem, we will find secant and tangent lines to this function. 3 2 2 y = g(x) a) (2 points) On the graph above, draw the secant line of g(x) between x = -2 and x = 5. Label this line with a large S. b) (2 points) Find the slope of the secant line you drew in Part (a). c) (3 points) Find an equation of the secant line you drew in Part (a). Equation: d) (2 points) On the graph above, draw the line tangent to g(x) at x=3. Label this line with a large T. e) (2 points) Evaluate g(4). g(4) f) (4 points) Use the definition of continuity to verify that g(x) is continuous at x = 4. (Do not use the graph for this part. We want to verify that the graph is displayed correctly.) g) (4 points) For x <4, find a formula for 8(x)-8(4) x-4 SIMPLIFY your result. h) (4 points) Evaluate lim g(x)-g(4) x-4- x-4 g(x)-g(4) x-4 lim g(x)-g(4) x++4- x-4 i) (4 points) For x > 4, find a formula for g(x) g(4) SIMPLIFY your result. x-4 j) (4 points) Evaluate lim g(x)-g(4) x+4+ x-4 g(x)-g(4) x-4 lim x-+4+ g(x)-g(4) x-4 k) (2 points) Use your answers from above to determine if g'(4) exists. Explain. 1) (3 points) Find an equation for the line tangent to g(x) at x = 4, if this line exists. If this tangent line does not exist, explain why. Equation: A piecewise defined function g is given by the formula -x+1 if x 4 g(x) if x > 4 A partial graph of this function is given below. In this problem, we will find secant and tangent lines to this function. 3 2 2 y = g(x) a) (2 points) On the graph above, draw the secant line of g(x) between x = -2 and x = 5. Label this line with a large S. b) (2 points) Find the slope of the secant line you drew in Part (a). c) (3 points) Find an equation of the secant line you drew in Part (a). Equation: d) (2 points) On the graph above, draw the line tangent to g(x) at x=3. Label this line with a large T. e) (2 points) Evaluate g(4). g(4) f) (4 points) Use the definition of continuity to verify that g(x) is continuous at x = 4. (Do not use the graph for this part. We want to verify that the graph is displayed correctly.) g) (4 points) For x <4, find a formula for 8(x)-8(4) x-4 SIMPLIFY your result. h) (4 points) Evaluate lim g(x)-g(4) x-4- x-4 g(x)-g(4) x-4 lim g(x)-g(4) x++4- x-4 i) (4 points) For x > 4, find a formula for g(x) g(4) SIMPLIFY your result. x-4 j) (4 points) Evaluate lim g(x)-g(4) x+4+ x-4 g(x)-g(4) x-4 lim x-+4+ g(x)-g(4) x-4 k) (2 points) Use your answers from above to determine if g'(4) exists. Explain. 1) (3 points) Find an equation for the line tangent to g(x) at x = 4, if this line exists. If this tangent line does not exist, explain why. Equation: