Given the function 𝑓(𝑥)=2𝑥2+3𝑥+1,Find the instantaneous rate of change when 𝑥=1 using the secant method. Bold text start[4 marks]Bold text EndFind the value of the derivative at 𝑥=1 using first principles.

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.

For the curve ݕ = 5ݔ − 2ݔଷ ,if ݔ increases at the rate of 2units/sec, then atݔ = 3the slope of the curve is changing at

The path of a baseball relative to the ground can be modelled by the function 𝑑(𝑡)=−𝑡2+8𝑡+1 where 𝑑(𝑡) represents the height of the ball in metres after 𝑡 seconds.Find the average rate of change of the ball between 1 and 3 seconds. Bold text start[4 marks]Bold text EndUsing the secant method, find the instantaneous rate of change at 2 seconds.

Determine the average rate of change of the function over theintervalb) Copy the graph, and draw a tangent line at the point whereDetermine the slope of the tangent line to estimate theinstantaneous rate of change at this point.2. Estimate the instantaneous rate of change of the function inquestion 1 at by determining the slope of a secant line fromthe point where to the point where Compare youranswer with your answer for question 1, part b).3. Use graphing technology to estimate the instantaneous rate of changeof the function in question 1 atPRACTISING4. Estimate the instantaneous rate of change of at thepoint (2, 21).f (x) 5 xx 2 4x 5 2.x 5 2.01.x 5 2x 5 2x 5 2.2 # x # 7.In SummaryKey Ideas• The methods that were previously used to calculate the average rate of changeand estimate the instantaneous rate of change can be used for rationalfunctions.• You cannot determine the average and instantaneous rates of change of arational function at a point where the graph is discontinuous (that is, wherethere is a hole or a vertical asymptote).Need to Know• The average rate of change of a rational function, on the intervalfrom is Graphically, this is equivalent to the slopeof the secant line that passes through the points and on thegraph of• The instantaneous rate of change of a rational function, at canbe approximated using the difference quotient and a very smallvalue of h. Graphically, this is equivalent to estimating the slope of the tangentline that passes through the point on the graph of• The instantaneous rate of change at a vertical asymptote is undefined. Theinstantaneous rates of change at points that are approaching a verticalasymptote become very large positive or very large negative values. Theinstantaneous rate of change near a horizontal asymptote approaches zero.y 5 f(x).(a, f(a))f(a 1 h) 2 f(a)hx 5 ay 5 f(x),y 5 f(x).(x2, y2)(x1, y1)f(x2) 2 f(x1)x2 2 x1 .x1 # x # x2y 5 f(x),01010–10–10yxy = x + 2——–-x – 1NEL 303Chapter 55.6

he tangent line at $$x=0 has been graphed, and its equation can be approximated by $$y=2.1x+3. Use the tangent to calculate the instantaneous rate of change at $$x=0.