Download Homework 8 - UC Davis Mathematics

Math 16A - Short Calculus
Homework 7
Exponential Functions, Higher Order Derivatives
Instructions: Show all steps in the solution. Compute the indicated derivatives of the following
functions.
1. Compute the derivative with respect to the independent variable of the following functions:
(a) f (x) = e6x
(b) g(t) = 3e12t
(c) h(s) = 2e4−3s
(d) y = ex
2
+x+1
(e) y = 5x
(f) y = 4 · π x
(g) y = 2e2x
2
−3x+4
(h) f (x) = esin(x)
√
(i) f (z) = ez − 1
1
(j) f (t) = 12e4t − e6t − t2
3
√
x
(k) f (x) = e
(l) f (,) = e6,−1 + ,2
2
(m) g(s) = 2s
−1
√
(n) f (x) = e
x−1+1
2. Compute the second derivative of the following functions:
(a) − cos(x)
(b) x4 − 3x3 + x + 1
(c) e4x
(d) ex+7
x2 − x + 5
2
(f) sin(3x + 1) − 2e4x + x
(e)
(g) − cos(1 − x) +
x2
−4
2
(h) 23x−1
3. This problem is meant to give us some intuition about how derivatives and polynomials react to each
other.
(a) Compute f 00 (x) for f (x) = x + 1.
(b) Compute f 000 (x) for f (x) = x2 + x + 1.
d4 3
(x + x2 + x + 1).
dx4
(d) Look at the number of derivatives you have taken and the powers on the function they acted on.
dn+1 n
Make a conjecture about what happens if we tried to compute
(x ).
dxn+1
(c) Compute
1
4. Find the equation of the tangent line to the given function through the given point.
(a) f (x) = sin(7x),
x=0
2
(b) g(x) = x − x + 4,
4x
(c) h(x) = 2e ,
x=0
x=0
3
(d) j(x) = x − 4x + 1,
2
(e) k(x) = −x − 3x + 1,
x=1
x=4
5. In this problem we will explore two new functions, called hyperbolic trigonometric functions. We will
see applications of them on later this quarter. The two hyperbolic trig functions are
sinh(x) =
ex − e−x
2
cosh(x) =
ex + e−x
2
(a) Determine what well known function is obtained from sinh(x) + cosh(x).
(b) Determine what well known function is obtained from cosh(x) − sinh(x).
(c) Compute the derivatives of both sinh(x) and cosh(x).
(d) What is the difference between the relationship of the derivatives of the hyperbolic trig functions
versus the relationship of the derivatives of the normal trig functions sin(x) and cos(x)?
(e) Show that the sum formula for sin(x) also works for sinh(x). In other words, show that
sinh(2x) = 2 sinh(x) cosh(x).
2