Using Different MethodsIn Exercises 47–50, find d w / d t (a) by using appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w = x 2 z + y + z , x = e t , y = t , z = t 2
Using Different MethodsIn Exercises 47–50, find d w / d t (a) by using appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w = x 2 z + y + z , x = e t , y = t , z = t 2
Solution Summary: The author explains how to calculate the derivative of w=x2z+y+z with respect to t using the chain rule.
Using Different MethodsIn Exercises 47–50, find
d
w
/
d
t
(a) by using appropriate Chain Rule and (b) by converting
w
to a function of
t
before differentiating.
w
=
x
2
z
+
y
+
z
,
x
=
e
t
,
y
=
t
,
z
=
t
2
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.