Problem 6 Define f(x)=x3+2x+1 for all x. Find the equation of the tangent line g
ID: 2983851 • Letter: P
Question
Problem 6
Define f(x)=x3+2x+1 for all x. Find the equation of the tangent line graph of f:R rightarrow R at the point (2,13). For m1 and m2 numbers, with m1 m2, define f(x) Prove that the function f:R rightarrow R is continuous but not differentiable at x=0. Use the definition of derivative to compute the derivative of the following functions at x=1: f(x)= for all x>0. F(x)=x3+2x for all x. f(x)=1/(1+x2) for all x. Evaluate the following limit s or determine that they do not exist: x2/x x2-1 x-1 x4-16/x-2 Let I and J be open intervals, and the functions f:I rightarrow R and h:J have the property that h(J) C I, so the composition f 0 h:J rightarrow R is defined. Show that X0 is J, h: J rightarrow R is continuous at X0, h(x) h(x0) if x x0, and f:I rightarrow f is differentiable at h(X0), then f(h(x))-f(h(x0))/h(x)-h(x0)=f'(h(x0)). Use Exercise 6 to show that f: R rightarrow R is differentiable at x0=1, these: f(1+h)-f(I)/h=f'(1) f( t)-f(I)/ t-1=f'(1) f(x2)-f(1)/ x2-1=f'(1) f(x2)-f(I)/x-1=2f'(1) f(x3)-f(I)/x-1=3f'(1). (Hint: For the last two limits, first make use of the difference of powers formula) For a natural number n 2,defineExplanation / Answer
3).
f(0-)=f(0+)=4 so continuous
f'(0-)=m1 and f'(0+)=m2 so non differentiable
4).
a). 1/2sqrt(2)
b).5
c).-1/2
5).
a).0
b).4
c).2
d).32
7).
a)by definition of a function to be derivative at h , f'(h) = [f(x+h)-f(x)]/h
b).Similarly as above
c).Similarly as above
d).Multiply and divide given equation by (x+1)
this will give you (x+1).f'(h)
putting h=1
=2f'(1)