24. Find the slope-point form of the equation of the tangent line to the graph o

Question

24. Find the slope-point form of the equation of the tangent line to the graph of 9 at the point (a, e^a ) . Differentiate the following functions. 35. The graph of y = x - e^x has one extreme point. Find its coordinates and decide whether it is a maximum or a minimum. (Use the second derivative test.) 36. Find the extreme points on the graph of y = x^2 e^x, and decide which one is a maximum and which one is a minimum. 37. Find the point, on the graph of y = (1 + x^2)ex where the tangent line is horizontal. 38. Show that the tangent line to the graph of y = 9 at the point (a, ea) is perpendicular to the tangent line to the graph of y = cx at the point (a, e^a).

Explanation / Answer

24) y= e^x

y ' = e^x   to find slope we will plug x= a

y' = e^a

Now slope point formula   y-y1=m(x-x1)

y-e^a=e^a(x-a) answer

38) y =e^x

y' =e^x    to find slope plug x= a we get

y ' = e^a    -----(1)

for 2nd case y = e^-x

y' = -e^-x

to find slope we plug x= a we   get

y ' = -e^-a = -1/e^a say ----(2)

Multiply both slopes   we got   e^a * - 1/e^a =-1

if product of two slopes =-1 it means perpendicular hence proved

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