2. Given the function f(x) = 2x + 3x + 1, a. Find the instantaneous rate...

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2. Given the function f(x) = 2x + 3x + 1, a. Find the instantaneous rate of change when x = 1 using the secant method. [4 marks] b. Find the value of the derivative at x = 1 using first principles. [4 marks] 3. Explain the difference between a secant line and a tangent line. How do they relate to the rate of change of a function? Include a sketch of each type of line in your solution. [6 marks] 4. The path of a baseball relative to the ground can be modelled by the function d(t) = -t + 8t + 1 where d(t) represents the height of the ball in metres after t seconds. a. Find the average rate of change of the ball between 1 and 3 seconds. [4 marks] b. Using the secant method, find the instantaneous rate of change at 2 seconds. [5 marks] 5. Evaluate lim (3x + 7x x2 6. Evaluate the following limits. x-16 [3 marks] a. lim X4 X4 b. lim x C. lim 0- 8c3_5x2+17 6x+2x-4x 49+h-7 h d. lim Y2 y -8 2y-7y+12 4 -2 e. lim 2+h h h0 16) and explain the meaning of this value in the context of our course. [5 marks] [3 marks] [4 marks] [3 marks] [3 marks] 7. Find the slope of the tangent line at point (-2, 2) on the curve f(x) = 2x + 3x. [4 marks] 8. Given the function f(x) = x 10x + 3, a. Find f(x) using first principles. [5 marks] b. Graph f (x) and f(x) on the same axes. [2 marks] c. What do you notice about f (x) when f(x) = 0? [2 marks] 2. Given the function f(x) = 2x + 3x + 1, a. Find the instantaneous rate of change when x = 1 using the secant method. [4 marks] b. Find the value of the derivative at x = 1 using first principles. [4 marks] 3. Explain the difference between a secant line and a tangent line. How do they relate to the rate of change of a function? Include a sketch of each type of line in your solution. [6 marks] 4. The path of a baseball relative to the ground can be modelled by the function d(t) = -t + 8t + 1 where d(t) represents the height of the ball in metres after t seconds. a. Find the average rate of change of the ball between 1 and 3 seconds. [4 marks] b. Using the secant method, find the instantaneous rate of change at 2 seconds. [5 marks] 5. Evaluate lim (3x + 7x x2 6. Evaluate the following limits. x-16 [3 marks] a. lim X4 X4 b. lim x C. lim 0- 8c3_5x2+17 6x+2x-4x 49+h-7 h d. lim Y2 y -8 2y-7y+12 4 -2 e. lim 2+h h h0 16) and explain the meaning of this value in the context of our course. [5 marks] [3 marks] [4 marks] [3 marks] [3 marks] 7. Find the slope of the tangent line at point (-2, 2) on the curve f(x) = 2x + 3x. [4 marks] 8. Given the function f(x) = x 10x + 3, a. Find f(x) using first principles. [5 marks] b. Graph f (x) and f(x) on the same axes. [2 marks] c. What do you notice about f (x) when f(x) = 0? [2 marks]