Question

1. Let u(x) and v(x) be functions such that

u(1)=2,u′(1)=3,v(1)=6,v′(1)=−1

If f(x)=u(x)v(x), what is f′(1). Explain how you arrive at your answer.

2. If f(x) is a function such that f(5)=9 and f′(5)=−4, what is the
equation of the tangent line to the graph of y=f(x) at the point
x=5? Explain how you arrive at your answer.

3. Find the equation of the tangent line to the function g(x)=xx−2 at the point (3,3). Explain how you arrive at your answer.

4. The position of an object moving along a coordinate line is given by s(t)=306−t. Find the object's speed and acceleration at time t=5. Explain how you arrive at your answer.

Answer #1

1). Consider the following function and point.
f(x) = x3 + x + 3; (−2,
−7)
(a) Find an equation of the tangent line to the graph of the
function at the given point.
y =
2) Consider the following function and point. See Example
10.
f(x) = (5x + 1)2; (0, 1)
(a) Find an equation of the tangent line to the graph of the
function at the given point.
y =

1. Let f be the function defined by f(x) = x
2 on the positive real numbers. Find the
equation of the line tangent to the graph of f at the point (3,
9).
2. Graph the reflection of the graph of f and the line tangent to
the graph of f at the point
(3, 9) about the line y = x.
I really need help on number 2!!!! It's urgent!

1. Let f(x)=(x^2+1)(2x-3)
Find the equation of the line tangent to the graph of f(x) at
x=3.
Find the value(s) of x where the tangent line is horizontal.
2. The total sales S of a video game t months after being
introduced is given by the function
S(t)=(5e^x)/(2+e^x )
Find S(10) and S'(10). What do these values represent in terms
of sales?
Use these results to estimate the total sales at t=11 months
after the games release.

f(x) =x2 -x
use f'(x)=lim h->0 f(x+h) - f(x)/h
find:
1. f '(x)
2. f '(2)
3. Find the equation of a tangent line to the given function at
x=2
4. f ' (-3)
5. Find the equation of a tangent line to the given function at
x=-3

Consider the function f (x) = x/(2x+1)*2 .
(i) Find the domain of this function. (Start by figuring out any
forbidden values!)
(ii) Use (i) to write the equation of the vertical asymptote for
this function.
(iii) Find the limits as x goes to positive and negative
infinity,
(iv) Find the derivative of this function.
(v) Find the coordinates at point A(..,…), where the
x-coordinate is 1. Use exact fractions, never a decimal
estimate.
(vi) Find the equation of the...

Let f(x) = x^3 - x
a) Find the equation of the secant line through (0,f(0)) and
(2,f(2))
b) State the Mean-Value Theorem and show that there is only one
number c in the interval that satisfies the conclusion of the
Mean-Value Theorem for the secant line in part a
c) Find the equation of the tangent line to the graph of f at point
(c,f(c)).
d) Graph the secant line in part (a) and the tangent line in part...

Consider the function F(x, y, z) =x2/2−
y3/3 + z6/6 − 1.
(a) Find the gradient vector ∇F.
(b) Find a scalar equation and a vector parametric form for the
tangent plane to the surface F(x, y, z) = 0 at the point (1, −1,
1).
(c) Let x = s + t, y = st and z = et^2 . Use the multivariable
chain rule to find ∂F/∂s . Write your answer in terms of s and
t.

Let u=[6,2 ], v=[3,3 ], and b=[4,1 ]. Find
(x⋅u+y⋅v-b)×2 u, where x,y are scalars.

1.) Find the equation of the tangent line to the graph of the
function f(x)=5x-4/2x+2 at the point where x=2
2.) Find the derivative: r(t)=(ln(t^3+1))^2

g (u, v) is a differentiable function and g (1,2) = 100, gu
(1,2) = 3, gv (1,2) = 7 are given. The function f is defined as f
(x, y, z) = g (xyz, x ^ 2, y^2z). Find the equation of the tangent
plane at the point (1,1,1) of the f (x, y, z) = 100 surface.

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