GroupHomework #6

Let NiMvLUkiZkc2IjYjSSJ4R0YmLUknYXJjY29zR0YmRic=.a) Find values of the constants a and b so that the lionear function NiMvLUkiTEc2IjYjSSJ4R0YmLCZJImFHRiYiIiIqJkkiYkdGJkYrRihGK0Yr has the properties L(0)=f(0) and L'(0)=f'(0).b) Find the value of the constant c for which the quadratric function NiMvLUkiUUc2IjYjSSJ4R0YmLCYtSSJMR0YmRiciIiIqJkkiY0dGJkYsKiRGKCIiI0YsRiw= has the properties Q(0)=f(0), Q'(0)=f'(0) and Q''(0)=f''(0).c) Find the value of the constant d for which the cubic function NiMvLUkiQ0c2IjYjSSJ4R0YmLCYtSSJRR0YmRiciIiIqJkkiZEdGJkYsKiRGKCIiJEYsRiw= has the properties C(0)=f(0), C'(0)=f'(0), C''(0)=f''(0), and C'''(0)=f'''(0).d) Plot f, L, Q, and C on the same axes over the interval [-4,4].e) Find an interval over which NiMyLUkkYWJzR0kqcHJvdGVjdGVkR0YmNiMsJi1JImZHNiI2I0kieEdGKyIiIi1JIkxHRitGLCEiIi1JJkZsb2F0R0YmNiRGLkYx.f) Find an interval over which NiMyLUkkYWJzR0kqcHJvdGVjdGVkR0YmNiMsJi1JImZHNiI2I0kieEdGKyIiIi1JIlFHRitGLCEiIi1JJkZsb2F0R0YmNiRGLkYx.g) Find an interval over which NiMyLUkkYWJzR0kqcHJvdGVjdGVkR0YmNiMsJi1JImZHNiI2I0kieEdGKyIiIi1JIkNHRitGLCEiIi1JJkZsb2F0R0YmNiRGLkYx.

Hyperbolic functions are similar to the trig functions. We define hyperbolic cosine, denoted cosh(x), as NiMvLUklY29zaEc2IjYjSSJ4R0YmKiYsJi1JJGV4cEdGJkYnIiIiLUYsNiMsJEYoISIiRi1GLSIiI0Yx. We also define hyperbolic sine, denoted sinh(x), as NiMvLUklc2luaEc2IjYjSSJ4R0YmKiYsJi1JJGV4cEdGJkYnIiIiLUYsNiMsJEYoISIiRjFGLSIiI0Yx. a) Use the definition of cosh(x) and sinh(x) to show that NiMvLUklZGlmZkdJKnByb3RlY3RlZEdGJjYkLUklY29zaEc2IjYjSSJ4R0YqRiwtSSVzaW5oR0YqRis= and NiMvLUklZGlmZkdJKnByb3RlY3RlZEdGJjYkLUklc2luaEc2IjYjSSJ4R0YqRiwtSSVjb3NoR0YqRis=. b) Use the definition of cosh(x) and sinh(x) to show that NiMvLCYqJC1JJWNvc2hHNiI2I0kieEdGKCIiIyIiIiokLUklc2luaEdGKEYpRishIiJGLA==. c) The inverse cosh(x) and sinh(x) are denoted by arccosh(x) and arcsinh(x). Use implicit differentiation to compute NiMtSSVkaWZmR0kqcHJvdGVjdGVkR0YlNiQtSShhcmNjb3NoRzYiNiNJInhHRilGKw== and NiMtSSVkaWZmR0kqcHJvdGVjdGVkR0YlNiQtSShhcmNzaW5oRzYiNiNJInhHRilGKw==. d) Use the definition of cosh(x) to verify that NiMvLUkoYXJjY29zaEc2IjYjSSJ4R0YmLUkjbG5HRiY2IywmRigiIiItSSVzcXJ0R0YmNiMsJiokRigiIiNGLUYtRi1GLQ==.e) Use the definition of arccosh(x) from part (d) to compute NiMtSSVkaWZmR0kqcHJvdGVjdGVkR0YlNiQtSShhcmNjb3NoRzYiNiNJInhHRilGKw==. Does this match your answer from part (c)? Explain.

For what values of a and b is (1,6) an inflection point of y = x^3+a*x^2+b*x+1.

Find equations of the lines tangent to NiMvLCwqJCUieEciIiMiIiIqJCUieUdGJ0YoKiYiIidGKEYmRighIiIqJiIiJUYoRipGKEYtIiM3Ri0iIiE= and containing the point (-4,3).

That point isn't on the graph is it?