MATH114 Lecture Notes - Lecture 14: Difference Quotient

• School: University of Alberta
• Department: Mathematics
• Course: MATH114
• Professor: Powell,Elizabeth

Document Summary

In point-slope formula, the tangent line can be given by: The derivative at x = a, denoted by (cid:1858)"(cid:4666)(cid:4667) is given by the limit as h approaches zero of the. Instantaneous rate of change at (cid:1876)= = derivative of f(x) at (cid:1876)= = (cid:1858) (cid:4666)(cid:4667)=(cid:1865)= lim (cid:2868)(cid:4666)+ (cid:4667) (cid:4666)(cid:4667) (cid:1877)=(cid:1858) (cid:4666)(cid:4667)(cid:4666)(cid:1876) (cid:4667)+(cid:1858)(cid:4666)(cid:4667) for f at (a, f(a)) For example, given(cid:1858)(cid:4666)(cid:1876)(cid:4667)= (cid:883)6(cid:1872)(cid:2870), find the equation of the tangent line to f at t = 1 seconds. =lim (cid:2868) (cid:4666) (cid:885)(cid:884) (cid:883)6 (cid:4667) thrown up at an initial velocity of +32 feet/second, from an initial height of 112 feet at 1 second. The velocity function is the derivative of the position function thus, (cid:4666)(cid:883)(cid:4667) = (cid:1858)"(cid:4666)(cid:883)(cid:4667): lim (cid:2868)(cid:4666)(cid:2869)+ (cid:4667) (cid:4666)(cid:2869)(cid:4667) Therefore, (cid:4666)(cid:883)(cid:4667) = (cid:882), and the ball reaches the top of the arc in one second. The derivative as a function can find the velocity function: